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Every rule and concept in Algebra 1, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (225)

$a^2+b^2=c^2$: In a right triangle, the two shorter sides (legs) $a$ and $b$ and the longest side (hypotenuse) $c$ satisfy \[ a^2+b^2=c^2.; Radicals › The Pythagorean Theorem
$n$ th Roots and Rational Exponents: The index does not have to be $2$ or $3$ .; Radicals › Going Deeper: Advanced Radicals
A First Look at Complex Roots: When $D=b^2-4ac<0$ , the quadratic formula asks for the square root of a negative number, which no real number provides.; Quadratics › Going Deeper: Advanced Quadratics
A Fraction Within a Fraction: A complex fraction has fractions in its numerator, its denominator, or both.; Rational Expressions › Complex Fractions (Intro)
A Game Plan: Real problems mix several rules.; Exponents › Simplifying Combined Expressions
A Peek at Determinants: Cramer's Rule: For a $2\times2$ system \[; Systems of Equations › Going Deeper: Advanced Systems
A Preview of Summation ( $\Sigma$ ) Notation: The Greek capital sigma $\Sigma$ is shorthand for “add up a list.” The expression \[ ∑_i=1^n a_i = a_1 + a_2 + a_3 + … + a_n \] reads: “start the counter $i$ at $1$ , plug it into the rule after the $\Sigma$ , and add the results up to $i = …; Foundations › Going Deeper: Advanced Foundations
Absolute Value as Distance: $|a - b|$: Beyond “distance from $0$ ,” absolute value measures the distance between any two numbers: \[ |a - b| = the distance between a and b on the number line.; Foundations › Going Deeper: Advanced Foundations
Absolute value means distance: $|x - 3| = 7$ says " $x$ is $7$ away from $3$ " — so $x = 10$ or $x = -4$ .; Core Ideas in Plain Terms › Going Deeper: The Quadratic Formula and Absolute Value
Add the Exponents: When you multiply powers with the same base, keep the base and add the exponents.; Exponents › Product of Powers
Adding & Subtracting Signed Numbers: Adding:; Foundations › Operations with Integers (Sign Rules)
Adding and Subtracting in Scientific Notation: For multiplying and dividing you handle the coefficients and the powers of $10$ separately.; Exponents › Going Deeper: Advanced Exponents
Almost like equations: Inequalities use $<, >, \le, \ge$ instead of $=$ .; Equations & Inequalities › Solving Inequalities
Average Speed Is a Harmonic Mean: For equal distances traveled at two speeds, the average speed is not the plain average of the speeds.; Rational Expressions › Going Deeper: Advanced Rational Expressions
Axes, Origin, and Ordered Pairs: The coordinate plane is formed by two number lines that cross at the origin $(0,0)$ .; Linear Equations & Functions › The Coordinate Plane
Balance: An equation says that two things are equal, joined by an equals sign, like $x + 3 = 7$ .; Equations & Inequalities › What Is an Equation?
Build a Line from a Slope and Any Point: If a line has slope $m$ and passes through $(x_1,y_1)$ , its equation is \[ y-y_1=m(x-x_1).; Linear Equations & Functions › Point-Slope Form: $y-y_1=m(x-x_1)$
Building a Perfect Square: Completing the square rewrites $x^2+bx$ as a perfect square by adding $\left(\dfrac{b}{2}\right)^2$ .; Quadratics › Solving by Completing the Square
Center: Mean, Median, Mode: Course Review › Data & Statistics
Circles and arrows: Graph the solution of a one-variable inequality on a number line:; Equations & Inequalities › Graphing on a Number Line
Clear the denominators: Multiply every term on both sides by the least common denominator (LCD).; Equations & Inequalities › Equations with Fractions & Decimals
Clear the Denominators: To solve an equation with rational expressions:; Rational Expressions › Solving Rational Equations
Closure & the Field Properties of the Reals: A set of numbers is closed under an operation if combining any two members always lands you back inside that set.; Foundations › Going Deeper: Advanced Foundations
Collinearity and the Area of a Triangle: Three points $A(x_1,y_1)$ , $B(x_2,y_2)$ , $C(x_3,y_3)$ form a triangle whose area is given by the shoelace formula \[ Area= $\tfrac{1}{2}$ x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2).; Linear Equations & Functions › Going Deeper: Advanced Linear Functions
Combine LIKE Radicals: Radicals are like when they have the same index and the same radicand.; Radicals › Adding & Subtracting Radicals
Combine Like Terms: Like terms have the exact same variable part (same variable, same exponent), such as $3x^2$ and $5x^2$ .; Polynomials › Adding Polynomials
Combining $\sqrt{a}\pm\sqrt{b}$ Cleverly: Sums and differences of unlike radicals will not merge, but you can still simplify products and squares of them.; Radicals › Going Deeper: Advanced Radicals
Comparing Powers of Different Bases: You cannot just eyeball which of $2^{40}$ or $3^{30}$ is larger.; Exponents › Going Deeper: Advanced Exponents
Comparing Slopes: Linear Equations & Functions › Parallel and Perpendicular Lines
Continued & Nested Complex Fractions: A continued fraction stacks fractions inside fractions many layers deep.; Rational Expressions › Going Deeper: Advanced Rational Expressions
Counting integer solutions of an inequality: Some problems ask not for the solution set but for how many integers lie in it.; Equations & Inequalities › Going Deeper: Advanced Equations & Inequalities
Definition: A rational expression is a ratio of two polynomials—one polynomial on top (the numerator) divided by another polynomial on the bottom (the denominator): \[ polynomialpolynomial, as $\frac{2x+1}{x-3}$ , x^2…; Rational Expressions › What Is a Rational Expression?
Definition and Standard Form: A quadratic function is one whose highest power of the variable is $2$ .; Quadratics › What Is a Quadratic?
Degree: The degree of a term is the exponent on its variable ( $-5x^3$ has degree $3$ ; a constant like $7=7x^0$ has degree $0$ ).; Polynomials › Degree, Leading Coefficient, and Standard Form
Denesting a Nested Radical: A radical inside a radical such as $\sqrt{7+4\sqrt3}$ can sometimes be denested into $\sqrt{a}+\sqrt{b}$ .; Radicals › Going Deeper: Advanced Radicals
Describing the shape: Data & Statistics › Displaying Data & Describing Shape
Distance from a Point to a Line: The distance from a point $(x_0,y_0)$ to the line $Ax+By+C=0$ is the length of the perpendicular segment from the point to the line: \[ d= Ax_0+By_0+C $\sqrt{A^2+B^2}$ .; Linear Equations & Functions › Going Deeper: Advanced Linear Functions
Distance from zero: $|x|$ means the distance of $x$ from $0$ , which is never negative.; Equations & Inequalities › Absolute Value Equations & Inequalities
Distance from Zero: The absolute value of a number is its distance from $0$ on the number line.; Foundations › Absolute Value
Distribute Each Term: Multiply each term of the binomial by each term of the trinomial (that is $2\times 3 = 6$ small products), then combine like terms and write in standard form.; Polynomials › Multiplying a Binomial by a Trinomial
Distribute the Exponent: An exponent on a product or a quotient applies to every factor inside.; Exponents › Power of a Product and Power of a Quotient
Distribute the Negative: To subtract, distribute the minus sign to every term of the second polynomial (this flips each sign), and then combine like terms.; Polynomials › Subtracting Polynomials
Efficient Order of Operations with Nested Grouping: When grouping symbols are nested — parentheses inside brackets inside braces, or a fraction bar acting as grouping above and below — always resolve the innermost group first and work outward.; Foundations › Going Deeper: Advanced Foundations
End Behavior from the Leading Term: For large $|x|$ , a polynomial behaves like its leading term alone—the highest-degree term dominates every other.; Polynomials › Going Deeper: Advanced Polynomials
Even Powers Come Out Clean: Since $\sqrt{x^2}=x$ (for $x\ge 0$ ), any even power under a square root leaves half its exponent outside: \[ x^6=x^3, x^10=x^5.; Radicals › Radicals with Variables
Extending the Special Products: $(a+b+c)^2$: The perfect-square pattern generalizes to three terms.; Polynomials › Going Deeper: Advanced Polynomials
Extracting One Coefficient Without Full Expansion: Sometimes you only need a single coefficient of a product.; Polynomials › Going Deeper: Advanced Polynomials
Factor Out the GCF: The GCF of several terms is the largest factor they all share.; Factoring › Greatest Common Factor (GCF)
Factor, Cancel, Multiply: To multiply, factor everything, cancel any factor common to a top and a bottom, then multiply the remaining tops together and the remaining bottoms together.; Rational Expressions › Multiplying Rational Expressions
Factor, Then Cancel Common Factors: To simplify a rational expression:; Rational Expressions › Simplifying Rational Expressions
Factoring as a Calculator Shortcut: The difference of squares works on numbers, not just variables.; Factoring › Going Deeper: Advanced Factoring
Factoring is un-multiplying: To factor means to rewrite an expression as a product.; Core Ideas in Plain Terms › Factoring and Quadratics
Factoring Strategy (in order): Course Review › Factoring
Find the LCD: When denominators differ, build a least common denominator (LCD):; Rational Expressions › Adding & Subtracting with UNLIKE Denominators
Find Two Numbers: To factor $x^2+bx+c$ , find two numbers that \[ multiply to c add to b.; Factoring › Trinomials $x^2+bx+c$ (Leading Coefficient $1$ )
FOIL: FOIL names the four products you make when multiplying two binomials: \[ F_First O_Outer I_Inner L_Last \] Then combine the tw…; Polynomials › Multiplying Two Binomials (FOIL)
Forms of a Linear Equation: Course Review › Linear Equations & Functions
Four ways to picture data: Data & Statistics › Displaying Data & Describing Shape
Fractions $\leftrightarrow$ Decimals: Foundations › Fractions
Gather the variables: Move all variable terms to one side and all number terms to the other by adding or subtracting.; Equations & Inequalities › Variables on Both Sides
Get zero on one side: For anything quadratic, move everything to one side first — the zero-product property only works against zero.; Core Ideas in Plain Terms › Problem-Solving Playbook
Graph of $y = ax^2 + bx + c$: The graph is a parabola.; Course Review › Quadratic Functions
Growth and Decay by a Percent Rate: The model $y = a\cdot b^{x}$ often comes from a percent change per step.; Exponents › Going Deeper: Advanced Exponents
How Elimination Works: Add or subtract the two equations so that one variable cancels.; Systems of Equations › Method 3: Solving by Elimination (Linear Combination)
How Graphing Works: Graph both lines on the same axes.; Systems of Equations › Method 1: Solving by Graphing
How Substitution Works: Solve one equation for one variable, then substitute that expression into the other equation.; Systems of Equations › Method 2: Solving by Substitution
Index of $3$: A cube root $\sqrt[3]{n}$ asks: what number cubed gives $n$ ?; Radicals › Cube Roots (A Brief Look)
Inequalities, Compound & Absolute Value: Solve inequalities like equations, with one rule: multiplying or dividing by a negative flips the inequality sign.; Course Review › Solving Equations & Inequalities
Inputs and Outputs: The domain is the set of all allowed inputs (the $x$ -values).; Linear Equations & Functions › Domain and Range
Isolate, Square, Check: To solve an equation with a variable under a radical:; Radicals › Solving Square-Root Equations
Line of Best Fit (Linear Regression): Real data rarely lands perfectly on one line.; Linear Equations & Functions › Going Deeper: Advanced Linear Functions
Linear Programming: Optimizing Over a Region: When a system of linear inequalities defines a feasible region, we often want to maximize (or minimize) a linear objective function such as $P = 3x + 4y$ .; Systems of Equations › Going Deeper: Advanced Systems
Literal Equations & Solution Counts: Literal equation: solve for one letter in terms of the others using the same isolation steps.; Course Review › Solving Equations & Inequalities
Literal equations that need factoring: When you solve a formula for a letter that appears in more than one term, you cannot simply divide — you must first collect every term containing that letter on one side, then factor the letter out, and only then divide by the leftover expression.; Equations & Inequalities › Going Deeper: Advanced Equations & Inequalities
Look at the coefficients before you start: Both methods always work, so the choice is about arithmetic, not correctness. Use substitution when a variable is already alone ( $y = \ldots$ ) or has a coefficient of $1$ or $-1$ — isolating it costs nothing and creates no fractions. Use elimination…; Systems of Equations › Choosing a Method
Method: With four terms, group them in pairs, factor the GCF from each pair, and then factor out the matching binomial.; Factoring › Factoring by Grouping (Four Terms)
Moving Freely Among the Three Forms: Every non-vertical line can be written in three equivalent ways.; Linear Equations & Functions › Going Deeper: Advanced Linear Functions
Multi-Step Equations: Goal: isolate the variable.; Course Review › Solving Equations & Inequalities
Multiply by the Reciprocal: To divide, flip the second fraction (take its reciprocal) and multiply.; Rational Expressions › Dividing Rational Expressions
Multiply Outsides, Multiply Insides: \[ (a $\sqrt{m}$ )(b $\sqrt{n}$ )=ab $\sqrt{mn}$ .; Radicals › Multiplying Radicals
Multiply the Exponents: When a power is raised to another power, keep the base and multiply the exponents.; Exponents › Power of a Power
Multiplying & Dividing Signed Numbers: The rule is the same for both operations:; Foundations › Operations with Integers (Sign Rules)
Multiplying & Special Products: FOIL for two binomials — First, Outer, Inner, Last: \[ (a+b)(c+d) = ac + ad + bc + bd.; Course Review › Polynomials
Multiplying and Dividing in Scientific Notation: Handle the coefficients and the powers of $10$ separately, using the product and quotient rules on the $10$ 's.; Exponents › Scientific Notation
Naming Polynomials by the Number of Terms: Polynomials › What Is a Polynomial?
Negative Exponents: A negative exponent means take the reciprocal (flip it), then use the positive exponent: \[ a^-n = 1a^n, 1a^-n = a^n.; Exponents › Zero and Negative Exponents
Nested absolute values: peel from the outside in: An expression like $\bigl||x-3| - 2\bigr|$ has one absolute value inside another.; Equations & Inequalities › Going Deeper: Advanced Equations & Inequalities
Nonlinear Preview: a Line Meets a Parabola: Not every system is two lines.; Systems of Equations › Going Deeper: Advanced Systems
Number of Solutions: Course Review › Systems of Equations & Inequalities
One equation, many answers; two equations, usually one: A single equation like $2x + y = 7$ has infinitely many solutions — pick any $x$ and $y$ follows.; Systems of Equations › What a System Is Actually Asking
One inverse operation: The variable has one operation attached to it.; Equations & Inequalities › One-Step Equations
One, None, or Infinitely Many: Two lines can meet in exactly three ways:; Systems of Equations › How Many Solutions? Three Possibilities
Operations & Solving: Course Review › Rational Expressions
Order of Operations (PEMDAS): Course Review › Foundations
Outlier fences: the $1.5 \times \text{IQR}$ rule: Earlier we named outliers informally.; Data & Statistics › Going Deeper: Advanced Statistics
Parallel, Perpendicular & Function Notation: Course Review › Linear Equations & Functions
Parameter equations: when does a solution exist?: A parameter is a letter that stands for a fixed but unknown number (often $a$ , $b$ , $k$ , $m$ ).; Equations & Inequalities › Going Deeper: Advanced Equations & Inequalities
Pascal's Triangle: The coefficients $\binom{n}{k}$ are exactly the rows of Pascal's triangle.; Polynomials › Going Deeper: Advanced Polynomials
Piecewise and Step Functions: A piecewise function uses different rules on different parts of the domain.; Linear Equations & Functions › Going Deeper: Advanced Linear Functions
Plotting paired data: A scatter plot graphs pairs of data $(x, y)$ as points.; Data & Statistics › Scatter Plots & Correlation
Polynomial Long Division (Preview): When a polynomial does not divide evenly, long division splits it into a quotient plus a remainder, mirroring long division of numbers.; Polynomials › Going Deeper: Advanced Polynomials
Preview: Asymptotes & Holes of Rational Functions: The graph of $y=\dfrac{P(x)}{Q(x)}$ is shaped by where the function blows up or flattens.; Rational Expressions › Going Deeper: Advanced Rational Expressions
Preview: The Factor Theorem: How do you factor something that is not a nice pattern, like $x^3-7x+6$ ?; Factoring › Going Deeper: Advanced Factoring
Principal Root: Every positive number really has two square roots: $5\times 5 = 25$ and $(-5)\times(-5)=25$ .; Radicals › Square Roots & Perfect Squares
Product Property of Radicals: For non-negative numbers $a$ and $b$ : \[ $\sqrt{ab}$ = $\sqrt{a}$ · $\sqrt{b}$ .; Radicals › Simplifying Square Roots (Product Property)
Properties & Like Terms: Course Review › Foundations
Quartics in Quadratic Form: An expression like $x^4+Bx^2+C$ is secretly a quadratic in disguise.; Factoring › Going Deeper: Advanced Factoring
Quotient Property & Rationalizing: \[ $\dfrac{a}{b}$ = $\sqrt{a}$ $\sqrt{b}$ (b>0).; Radicals › Dividing & Rationalizing
Range, quartiles, and IQR: A measure of spread tells how spread out the data is.; Data & Statistics › Measures of Spread
Rational Exponents: the Bridge to Radicals: A fractional exponent is a compact way to write a root.; Exponents › Going Deeper: Advanced Exponents
Rationalizing & Pythagorean Theorem: Rationalize a denominator by removing the root from the bottom: \[ 1 $\sqrt{2}$ = 1 $\sqrt{2}$ · $\sqrt{2}$ $\sqrt{2}$ = $\sqrt{2}$ 2.; Course Review › Radicals
Rationalizing with a Conjugate: When a denominator is a sum or difference of terms involving radicals, one factor is not enough.; Radicals › Going Deeper: Advanced Radicals
Read the Slope and Intercept Directly: In slope-intercept form \[ y=mx+b, \] the number $m$ is the slope and $b$ is the $y$ -intercept (the $y$ -value where the line crosses the $y$ -axis, at the point $(0,b)$ ).; Linear Equations & Functions › Slope-Intercept Form: $y=mx+b$
Reading a proof: Do not read a proof the way you read a rule.; Rational Expressions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Data & Statistics › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Foundations › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Equations & Inequalities › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Linear Equations & Functions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Systems of Equations › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Exponents › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Polynomials › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Factoring › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Quadratics › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Radicals › Why It Works
Reading a two-way table: A two-way frequency table sorts data by two categories at once (rows and columns).; Data & Statistics › Two-Way Frequency Tables
Reading Off the Sum and Product of the Solutions: If a quadratic factors as $x^2+bx+c=(x-r)(x-s)$ , then multiplying out shows \[ x^2+bx+c=x^2-(r+s)x+rs.; Factoring › Going Deeper: Advanced Factoring
Reading the Number Line: The number line places every real number in order from small (left) to large (right).; Foundations › The Number Line & Ordering Numbers
Reading the Vertex Directly: The vertex form of a quadratic is \[ y = a(x-h)^2 + k, \] where the vertex is $(h,k)$ and $x=h$ is the axis of symmetry.; Quadratics › Vertex Form
Real Number Families: Every number in Algebra 1 is a real number, sorted into nested sets:; Course Review › Foundations
Rearranging formulas: A literal equation has several letters.; Equations & Inequalities › Literal Equations (Solving for a Variable)
Recovering $a^2+b^2$ and $a^3+b^3$ from $a+b$ and $ab$: The sum $a+b$ and product $ab$ contain enough information to rebuild higher symmetric expressions.; Polynomials › Going Deeper: Advanced Polynomials
Repeated Multiplication: An exponent is a shortcut for repeated multiplication.; Exponents › What an Exponent Means
Residuals: how wrong is a prediction?: A residual measures the error at a single data point: \[ residual = actual y - predicted y.; Data & Statistics › Going Deeper: Advanced Statistics
Same Bottom? Combine the Tops: When the denominators are already the same, add or subtract the numerators and keep the common denominator.; Rational Expressions › Adding & Subtracting with LIKE Denominators
Scatter Plots & Line of Best Fit: A scatter plot shows paired data to reveal a relationship (correlation):; Course Review › Data & Statistics
Scientific Notation: A number written as $a \times 10^n$ where $1 \le a < 10$ and $n$ is an integer.; Course Review › Exponents & Exponent Rules
Shading the Solution Region: A system of inequalities uses $<,\,>,\,\le,\,\ge$ instead of $=$ .; Systems of Equations › Systems of Linear Inequalities
Shape, Direction, and Key Points: The graph of $y=ax^2+bx+c$ is a smooth U-shaped curve called a parabola.; Quadratics › The Parabola: The Graph of a Quadratic
Simplify & Excluded Values: A rational expression is a fraction of polynomials.; Course Review › Rational Expressions
Simplify first: Before you isolate the variable, clean up each side: use the distributive property to remove parentheses, then combine like terms.; Equations & Inequalities › Multi-Step Equations
Simplifying & Operating: A radical is simplified when no perfect-square factor remains under the root.; Course Review › Radicals
Simplifying Fractions: A fraction is in lowest terms when the numerator and denominator share no common factor except $1$ .; Foundations › Fractions
Slope: Slope $m$ measures steepness (rise over run): \[ m = $\frac{y_2 - y_1}{x_2 - x_1}$ = riserun.; Course Review › Linear Equations & Functions
Slope = Rise over Run: Slope $m$ measures steepness: how much $y$ changes for each unit that $x$ increases.; Linear Equations & Functions › Rate of Change and Slope
Slope is steepness: rise over run: The slope of a line is how much it goes up for each step right — "rise over run." In $y = mx + b$ , the $m$ is that slope and $b$ is where the line crosses the $y$ -axis.; Core Ideas in Plain Terms › Lines and Slope
Solving Exponential Equations by Matching Bases: If two powers with the same base are equal, then their exponents must be equal: \[ b^m = b^n m = n (b>0, b≠ 1).; Exponents › Going Deeper: Advanced Exponents
Special Lines: Linear Equations & Functions › Horizontal and Vertical Lines
Spotting the case without solving: Write both equations as $ax + by = c$ .; Systems of Equations › No Solution, and Infinitely Many
Spread: Range & IQR: Course Review › Data & Statistics
Standard Form and Intercepts: Standard form is $Ax+By=C$ , where $A$ , $B$ , $C$ are integers.; Linear Equations & Functions › Standard Form: $Ax+By=C$
Standard Form and Leading Coefficient: A polynomial is in standard form when its terms are written in descending order of degree (highest power first).; Polynomials › Degree, Leading Coefficient, and Standard Form
Substitute and Simplify: To evaluate a polynomial for a given value, replace every variable with that value (use parentheses!), then follow the order of operations.; Polynomials › Evaluating a Polynomial
Substitute, then Simplify: To evaluate an algebraic expression, replace each variable with its given value (use parentheses!) and then follow PEMDAS.; Foundations › Evaluating Expressions by Substitution
Subtract the Exponents: When you divide powers with the same base, keep the base and subtract the bottom exponent from the top.; Exponents › Quotient of Powers
Sum and Difference of Cubes: Two more patterns join the difference of squares.; Factoring › Going Deeper: Advanced Factoring
Sum and Product of the Roots (Vieta's Formulas): If $r_1$ and $r_2$ are the two roots of $ax^2+bx+c=0$ , then \[ r_1+r_2 = - $\frac{b}{a}$ , r_1 r_2 = $\frac{c}{a}$ .; Quadratics › Going Deeper: Advanced Quadratics
Sums of absolute values as distances: Read $|x - a|$ as “the distance between $x$ and $a$ on the number line.” Then $|x-a| + |x-b|$ is the total distance from $x$ to the two fixed points $a$ and $b$ .; Equations & Inequalities › Going Deeper: Advanced Equations & Inequalities
Symmetric Systems: the $x+y$ and $xy$ Trick: A system is symmetric if swapping $x$ and $y$ leaves it unchanged (for example, using $x+y$ , $xy$ , or $x^2+y^2$ ).; Systems of Equations › Going Deeper: Advanced Systems
Systems in Three Variables: A three-variable system has three equations in $x$ , $y$ , and $z$ .; Systems of Equations › Going Deeper: Advanced Systems
Systems With a Parameter: Sometimes a coefficient is a letter (a parameter) and we ask: for which values does the system have one, none, or infinitely many solutions?; Systems of Equations › Going Deeper: Advanced Systems
Take the Square Root of Both Sides: When a quadratic has no plain $x$ term, isolate the squared quantity and take the square root of both sides—remembering both signs: \[ x^2 = k ⇒ x = ± $\sqrt{k}$ , (x-h)^2 = k ⇒ x-h = ± $\sqrt{k}$ .; Quadratics › Solving by the Square-Root Method
Telescoping Sums via Partial Fractions: Some long sums of rational terms collapse.; Rational Expressions › Going Deeper: Advanced Rational Expressions
The “ac” / Grouping Method: For $ax^2+bx+c$ :; Factoring › Trinomials $ax^2+bx+c$ (Leading Coefficient $\neq 1$ )
The $x+\dfrac{1}{x}$ Family: A classic trick: if you know the value of $x+\dfrac{1}{x}$ , you can find $x^2+\dfrac{1}{x^2}$ and $x^3+\dfrac{1}{x^3}$ without ever solving for $x$ .; Rational Expressions › Going Deeper: Advanced Rational Expressions
The Add-and-Subtract-a-Square Trick (Sophie Germain): A sum of squares usually will not factor — but sometimes we can create a difference of squares by adding and subtracting a middle term.; Factoring › Going Deeper: Advanced Factoring
The Big Idea: A system of equations is two (or more) equations that we want to be true at the same time.; Systems of Equations › What Is a System of Equations?
The Big Idea: Factoring means writing an expression as a product (things multiplied together).; Factoring › What Factoring Is (and Why It Helps)
The Big Idea: A square root of a number $n$ is a value that, when multiplied by itself, gives $n$ .; Radicals › Square Roots & Perfect Squares
The Binomial Theorem: FOIL and repeated distribution get slow for high powers.; Polynomials › Going Deeper: Advanced Polynomials
The Building Blocks: A term is a number, a variable, or a number multiplied by variables raised to whole-number powers.; Polynomials › What Is a Polynomial?
The Complete Rule Sheet: For any nonzero base(s) $a$ , $b$ and integers $m$ , $n$ :; Exponents › All the Exponent Rules at a Glance
The correlation coefficient $r$: The words “strong” and “weak” can be made precise with the correlation coefficient $r$ , a number between $-1$ and $1$ that measures how tightly points cluster around a straight line.; Data & Statistics › Going Deeper: Advanced Statistics
The Discriminant: The discriminant is $\;D = b^2 - 4ac$ (under the radical).; Course Review › Quadratic Functions
The Discriminant and the Nature of the Roots: For a quadratic with rational coefficients, the discriminant $D=b^2-4ac$ tells you not just how many roots, but what kind:; Quadratics › Going Deeper: Advanced Quadratics
The Distance Formula Is the Pythagorean Theorem: Place two points $(x_1,y_1)$ and $(x_2,y_2)$ on the plane.; Radicals › Going Deeper: Advanced Radicals
The Distributive Property: Multiply the monomial by each term inside the parentheses.; Polynomials › Multiplying a Monomial by a Polynomial
The Exponent Rules: For any nonzero base:; Course Review › Exponents & Exponent Rules
The five-number summary: The five-number summary lists, in order: \[ minimum, Q1, median, Q3, maximum.; Data & Statistics › Five-Number Summary & Box Plots
The Form: Scientific notation writes a number as \[ c × 10^n, where 1 ≤ c < 10 and n is an integer. \] The factor $c$ (the coefficient) has exactly one nonzero digit before the decimal point.; Exponents › Scientific Notation
The Formula That Always Works: For any quadratic equation $ax^2+bx+c=0$ , \[ x=-b± $\sqrt{b^2-4ac}$ 2a.; Quadratics › The Quadratic Formula
The Four Operations: Foundations › Fractions
The Line-Tangent-to-Parabola Condition: To find where a line $y=mx+d$ meets a parabola $y=ax^2+bx+c$ , set them equal and collect everything on one side to get a single quadratic in $x$ .; Quadratics › Going Deeper: Advanced Quadratics
The Logarithm: Undoing an Exponent (Preview): Exponentiation asks “what do I get when I raise $b$ to the power $x$ ?” The logarithm asks the reverse: “what power of $b$ gives me $y$ ?” They are inverse operations: \[ b^x = y _b y = x (b>0, b≠ 1).; Exponents › Going Deeper: Advanced Exponents
The method does not know whether a symbol is a number: Elimination multiplies, adds and divides.; Systems of Equations › When the Coefficients Are Letters
The Named Properties: Foundations › Properties of Operations
The Number Families: Every number you will use in Algebra 1 is a real number.; Foundations › Types of Real Numbers
The Number Under the Radical: The discriminant is the expression inside the square root of the quadratic formula: \[ D = b^2 - 4ac.; Quadratics › The Discriminant
The Order: Simplify in this order:; Foundations › Order of Operations (PEMDAS)
The Pattern: \[ a^2-b^2=(a+b)(a-b).; Factoring › Difference of Squares
The Pattern $y = a\cdot b^{x}$: Many real situations follow \[ y = a· b^x, \] where $a$ is the starting amount (the value when $x=0$ ) and $b$ is the growth factor.; Exponents › Exponential Growth vs. Decay (Intro)
The Patterns: Factoring › Perfect-Square Trinomials
The procedure: 1.; Systems of Equations › Substitution, Step by Step
The Property: If a product equals zero, then at least one factor must be zero: \[ if A· B=0, A=0 or B=0.; Factoring › Zero Product Property (Solving by Factoring)
The quadratic formula, demystified: Completing the square once, in general, on $ax^2 + bx + c = 0$ gives $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ — so the formula is not magic, it is the same move you already know, done once for every equation.; Core Ideas in Plain Terms › Going Deeper: The Quadratic Formula and Absolute Value
The three averages: A measure of center is a single number that describes the “middle” of a data set.; Data & Statistics › Measures of Center
The Work Formula: If one worker (or pipe) finishes a job in $a$ units of time and another in $b$ units, each does a fraction of the job per unit of time: $\frac{1}{a}$ and $\frac{1}{b}$ .; Rational Expressions › Applications: Work & Rate Problems
The Zero Exponent: Any nonzero base raised to the $0$ power equals $1$ : \[ a^0 = 1 (a ≠ 0).; Exponents › Zero and Negative Exponents
The Zero Product Property: If a product of two factors equals zero, then at least one factor must be zero: \[ if A· B = 0, then A=0 or B=0.; Quadratics › Solving by Factoring (Zero Product Property)
Three Common Situations: Linear Equations & Functions › Writing the Equation of a Line
Three possible outcomes, and how each one announces itself: One solution.; Systems of Equations › What a System Is Actually Asking
Three Solution Methods: A solution $(x,y)$ satisfies both equations — the point where the graphs cross.; Course Review › Systems of Equations & Inequalities
Trapped Between Two Integers: If a number is not a perfect square, its square root is irrational (a never-ending, non-repeating decimal).; Radicals › Estimating Irrational Square Roots
Trend lines and predictions: A line of best fit (or trend line) is a straight line that comes as close as possible to the points of a scatter plot with a linear pattern.; Data & Statistics › Line of Best Fit: Slope & Intercept
Turning Words into a Quadratic: Many real situations are modeled by quadratics:; Quadratics › Applications
Turning Words into a System: Many word problems hide a $2\times2$ system.; Systems of Equations › Applications: Word Problems
Two conditions at once: A compound inequality joins two inequalities.; Equations & Inequalities › Compound Inequalities (And / Or)
Two equations, two unknowns: A system is two equations that must both be true.; Core Ideas in Plain Terms › Systems of Equations
Two Patterns Worth Memorizing: Polynomials › Special Products
Two unknowns need two facts: A word problem becomes a system as soon as you name the unknowns and find two independent statements about them.; Systems of Equations › Systems in Words
Undo in reverse order: When two operations act on the variable, undo them in the reverse of the order of operations: first undo addition/subtraction, then undo multiplication/division.; Equations & Inequalities › Two-Step Equations
Units-Digit Cycles: The units digit (last digit) of a power repeats in a short cycle as the exponent grows, because only the previous units digit affects the next one.; Exponents › Going Deeper: Advanced Exponents
Variance & standard deviation: spread around the mean: The range and IQR describe spread using just a few values.; Data & Statistics › Going Deeper: Advanced Statistics
Vertex Form for Maximum/Minimum Optimization: Rewriting $y=ax^2+bx+c$ as $y=a(x-h)^2+k$ turns optimization into reading off the vertex: the extreme value of $y$ is exactly $k$ , reached at $x=h$ .; Quadratics › Going Deeper: Advanced Quadratics
Vocabulary & Add/Subtract: A polynomial is a sum of terms $ax^n$ (whole-number exponents).; Course Review › Polynomials
Ways to Solve a Quadratic: Course Review › Quadratic Functions
Weighted averages: An ordinary mean treats every value equally.; Data & Statistics › Going Deeper: Advanced Statistics
What Is a Function?: A function is a rule that assigns to each input exactly one output.; Linear Equations & Functions › Functions and Function Notation
What Makes Terms “Like”: Like terms have exactly the same variable part (same letters with the same exponents).; Foundations › Combining Like Terms
When the variable disappears: Sometimes, while solving, all the variable terms cancel out.; Equations & Inequalities › Special Cases: No Solution or Infinitely Many
Where the Quadratic Formula Comes From: The quadratic formula is not magic—it is completing the square done once and for all on the general equation $ax^2+bx+c=0$ .; Quadratics › Going Deeper: Advanced Quadratics
Which center is “best”?: Data & Statistics › Measures of Center
Which Method Should I Use?: Systems of Equations › Choosing the Best Method
Why the Denominator Matters: Division by zero is undefined.; Rational Expressions › Excluded Values (Domain Restrictions)
Why the Slope Rules Are True (Two Short Proofs): Parallel lines have equal slopes.; Linear Equations & Functions › Going Deeper: Advanced Linear Functions
Zero Product Property: If a product equals zero, at least one factor is zero: \[ if (x - r)(x - s) = 0, then x = r or x = s.; Course Review › Factoring