Core Ideas in Plain Terms

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. A bigger $m$ is steeper; a negative $m$ goes downhill.

To factor means to rewrite an expression as a product. For $x^2 + 5x + 6$, look for two numbers that multiply to $6$ and add to $5$ — that is $2$ and $3$ — so it factors as $(x+2)(x+3)$. Factoring reveals the roots through the zero-product property: if $ab = 0$ then $a = 0$ or $b = 0$. And factoring itself is the distributive property run in reverse — expanding back gives $(x+2)(x+3) = x^2 + 3x + 2x + 6$.

A system is two equations that must both be true. Solve by substitution (solve one for a variable and plug in) or elimination (add or subtract the equations to cancel a variable). The solution is the point where the two lines cross.

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. The part under the root, $b^2 - 4ac$, tells you how many real solutions there are: positive $\to$ two, zero $\to$ one, negative $\to$ none.

$|x - 3| = 7$ says "$x$ is $7$ away from $3$" — so $x = 10$ or $x = -4$. Reading absolute-value equations as distances turns them from a rule to memorize into a picture on the number line.

For anything quadratic, move everything to one side first — the zero-product property only works against zero. Then factor (or use the quadratic formula), and check by substituting back.