Quadratics

A quadratic function is one whose highest power of the variable is $2$. Its standard form is

A quadratic equation sets a quadratic equal to zero: $ax^2+bx+c=0$. The numbers $a$, $b$, and $c$ are the coefficients:

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• $a$ is the coefficient of $x^2$ (it can never be $0$, or the $x^2$ would vanish);
• $b$ is the coefficient of $x$;
• $c$ is the constant term.

The graph of $y=ax^2+bx+c$ is a smooth U-shaped curve called a parabola.

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• Direction: if $a>0$ the parabola opens up (holds water, has a lowest point); if $a<0$ it opens down (has a highest point).
• Vertex: the turning point. Its $x$-coordinate is $x=-\dfrac{b}{2a}$; find the $y$-coordinate by substituting that $x$ back in.
• Axis of symmetry: the vertical line $x=-\dfrac{b}{2a}$ through the vertex; the two halves mirror each other.
• Minimum vs. maximum: an up-parabola's vertex is a minimum; a down-parabola's vertex is a maximum.
• $y$-intercept: where the curve crosses the $y$-axis; set $x=0$ to get the point $(0,c)$.
• $x$-intercepts (roots / zeros): where the curve crosses the $x$-axis; set $y=0$ and solve $ax^2+bx+c=0$. A parabola can have two, one, or no $x$-intercepts.

The vertex is the turning point.

If a product of two factors equals zero, then at least one factor must be zero:

To use it, first make one side $0$, factor the quadratic, set each factor equal to $0$, and solve.

When a quadratic has no plain $x$ term, isolate the squared quantity and take the square root of both sides---remembering both signs:

This works whenever the variable appears only inside a square.

Completing the square rewrites $x^2+bx$ as a perfect square by adding $\left(\dfrac{b}{2}\right)^2$. The steps for $x^2+bx+c=0$:

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• Move the constant to the right side.
• Add $\left(\frac{b}{2}\right)^2$ to both sides.
• Write the left side as $\left(x+\frac{b}{2}\right)^2$.
• Take the square root of both sides ($\pm$!) and solve for $x$.

If $a\neq 1$, divide every term by $a$ first.

For any quadratic equation $ax^2+bx+c=0$,

It comes from completing the square on the general equation, so it solves every quadratic---even ones that will not factor nicely.

The discriminant is the expression inside the square root of the quadratic formula:

Its sign tells you how many real solutions the equation has, before you ever finish solving:

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• $D>0$: two different real solutions (parabola crosses the $x$-axis twice);
• $D=0$: one real solution, a repeated root (parabola just touches the $x$-axis);
• $D<0$: no real solutions (parabola never reaches the $x$-axis).

The vertex form of a quadratic is

where the vertex is $(h,k)$ and $x=h$ is the axis of symmetry. As always, $a>0$ opens up (minimum at the vertex) and $a<0$ opens down (maximum). Watch the sign: the form uses $(x-h)$, so $y=(x+1)^2-5$ has $h=-1$.

The vertex is the turning point.

Many real situations are modeled by quadratics:

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• Projectile height: an object thrown or launched has height $h=-16t^2+v_0 t + h_0$ (feet, seconds), where $v_0$ is the starting speed and $h_0$ the starting height. Set $h=0$ to find when it lands; use $t=-\frac{b}{2a}$ for the time of greatest height.
• Area: length $\times$ width often produces an $x^2$ term.

Solve, then keep only answers that make sense (a length or a time cannot be negative).

The quadratic formula is not magic---it is completing the square done once and for all on the general equation $ax^2+bx+c=0$. Because the derivation never uses specific numbers, the result works for every quadratic. The key move is the same as before: turn $x^2+\frac{b}{a}x$ into a perfect square by adding $\left(\frac{b}{2a}\right)^2$ to both sides. Master this derivation and you never have to memorize the formula blindly again.

The vertex is the turning point.

For a quadratic with rational coefficients, the discriminant $D=b^2-4ac$ tells you not just how many roots, but what kind:

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• $D>0$ and a perfect square: two rational roots (the quadratic factors over the rationals);
• $D>0$ but not a perfect square: two irrational roots, a conjugate pair $\frac{-b}{2a}\pm\frac{\sqrt{D}}{2a}$;
• $D=0$: one repeated rational root (a “double root”); the parabola is tangent to the $x$-axis;
• $D<0$: no real roots---two complex conjugate roots (see below).

If $r_1$ and $r_2$ are the two roots of $ax^2+bx+c=0$, then

You can see why by expanding $a(x-r_1)(x-r_2)=ax^2-a(r_1+r_2)x+a\,r_1 r_2$ and matching coefficients. For a monic quadratic ($a=1$), this reads especially cleanly:

These give a fast check on any answer, and let you build a quadratic straight from conditions on its roots.

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 $x=h$.

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• If $a>0$, the vertex is the lowest point, so $k$ is the minimum value.
• If $a<0$, the vertex is the highest point, so $k$ is the maximum value.

This is the standard way to answer “what is the largest area / smallest cost / greatest height?”

When $D=b^2-4ac<0$, the quadratic formula asks for the square root of a negative number, which no real number provides. We introduce the imaginary unit

Then $\sqrt{-16}=\sqrt{16}\,\sqrt{-1}=4i$, and the formula still works. The two roots come out as a complex conjugate pair $p\pm qi$: same real part $p=-\frac{b}{2a}$, opposite imaginary parts. Graphically, the parabola simply never crosses the $x$-axis.

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 $x$. The discriminant of that quadratic decides the geometry:

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• $D>0$: the line cuts the parabola at two points (a secant);
• $D=0$: the line just touches it at one point---it is tangent;
• $D<0$: the line misses the parabola entirely.

So “tangent” is precisely the condition $D=0$: one repeated intersection.