Chapter 09 of 10

Quadratics

How do equations with a squared variable behave?

A quadratic function has degree two and graphs as a parabola. Its vertex, axis of symmetry, intercepts, and direction of opening describe the graph. The form of the equation reveals different features, while factoring, square roots, completing the square, and the quadratic formula provide different routes to its zeros.

The big ideas

Understand these before you worry about speed.

Standard form describes the family

In standard form, a cannot be zero. Its sign tells whether the parabola opens up or down, while its size controls how narrow or wide the graph is. The constant c is also the y-intercept because substituting x = 0 gives y = c.

ExampleIn

\displaystyle y=-2x^2+8x+3

, the graph opens down and crosses the

\displaystyle y

-axis at

\displaystyle (0,3)

Vertex form shows the turning point

Vertex form names the vertex directly as (h, k) and the axis of symmetry as x = h. Be careful with the sign inside the parentheses: x − 4 means h = 4, while x + 4 means h = −4.

Example

\displaystyle y=3(x-2)^2-5

has vertex

\displaystyle (2,-5)

, axis

\displaystyle x=2

, and opens upward.

Factored form shows the zeros

When a quadratic is written as a product equal to zero, the zero-product property says at least one factor must be zero. Those solutions are the x-intercepts of the graph.

Example

\displaystyle y=(x-3)(x+1)

has zeros

\displaystyle x=3

and

\displaystyle x=-1

Use square roots when a square is isolated

If the equation has the form (x − h)² = d, take both the positive and negative square root. The ± is essential because two different numbers can have the same square. If d is negative, there are no real solutions.

Example

\displaystyle (x-4)^2=9

gives

\displaystyle x-4=\pm3

, so

\displaystyle x=1

\displaystyle x=7

Completing the square creates vertex form

Move the constant, divide by the leading coefficient when necessary, then add the square of half the x-coefficient to both sides. This creates a perfect-square trinomial and explains where vertex form and the quadratic formula come from.

Example

\displaystyle x^2+6x=7

becomes

\displaystyle x^2+6x+9=16

, then

\displaystyle (x+3)^2=16

The quadratic formula always works

For any equation in standard form ax² + bx + c = 0, identify a, b, and c with their signs and substitute them carefully. The entire numerator is divided by 2a, and the ± produces the two possible roots.

ExampleFor

\displaystyle 2x^2-3x-2=0

, use

\displaystyle a=2

\displaystyle b=-3

, and

\displaystyle c=-2

\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

The discriminant predicts the kind of roots

The expression under the radical in the quadratic formula is the discriminant. A positive value gives two distinct real roots, zero gives one repeated real root, and a negative value gives two complex roots and no x-intercepts.

ExampleFor

\displaystyle x^2+2x+5=0

\displaystyle b^2-4ac=2^2-4(1)(5)=-16

, so there are no real roots.

Worked example

Follow the reason for each step—not just the symbols.

Problem

Solve $\displaystyle 2x^2-3x-2=0$ with the quadratic formula.

Write the equation in $\displaystyle ax^2+bx+c=0$ form and identify $\displaystyle a=2$ , $\displaystyle b=-3$ , and $\displaystyle c=-2$ .
Substitute into $\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ . Keep the sign of $\displaystyle b$ .
Simplify the discriminant: $\displaystyle b^2-4ac=(-3)^2-4(2)(-2)=9+16=25$ .
Now $\displaystyle x=\frac{3\pm\sqrt{25}}{4}=\frac{3\pm5}{4}$ .
Evaluate both branches: $\displaystyle x=\frac{3+5}{4}=2$ or $\displaystyle x=\frac{3-5}{4}=-\frac{1}{2}$ .

Answer $\displaystyle x=2$ or $\displaystyle x=-\frac{1}{2}$

Try it yourself

Work on paper first. Open each answer only when you are ready to check.

For $\displaystyle y=-3(x-4)^2+2$ , state the vertex, axis of symmetry, and direction of opening.

Check answer

Vertex $\displaystyle (4,2)$ , axis $\displaystyle x=4$ , opens downward.

Find the zeros of $\displaystyle y=(x+2)(x-5)$ .

Check answer

$\displaystyle x=-2$ and $\displaystyle x=5$

Solve $\displaystyle (x+1)^2=25$ using square roots.

Check answer

$\displaystyle x=4$ or $\displaystyle x=-6$

Solve $\displaystyle x^2=20$ exactly.

Check answer

$\displaystyle x=\pm2\sqrt5$

Complete the square: $\displaystyle x^2+8x=9$ .

Check answer

$\displaystyle (x+4)^2=25$ , so $\displaystyle x=1$ or $\displaystyle x=-9$ .

Solve $\displaystyle x^2-7x+12=0$ by factoring.

Check answer

$\displaystyle x=3$ or $\displaystyle x=4$

Find the discriminant of $\displaystyle x^2+6x+9=0$ and classify the roots.

Check answer

$\displaystyle 0$ ; one repeated real root.

Use the quadratic formula to solve $\displaystyle 3x^2+x-2=0$ .

Check answer

$\displaystyle x=\frac{2}{3}$ or $\displaystyle x=-1$

Practice and tools

Use the resource that matches what you need next.

Calculator

Quadratics

The big ideas

Standard form describes the family

Vertex form shows the turning point

Factored form shows the zeros

Use square roots when a square is isolated

Completing the square creates vertex form

The quadratic formula always works

The discriminant predicts the kind of roots

Worked example

Solve $\displaystyle 2x^2-3x-2=0$ with the quadratic formula.

Try it yourself

Practice and tools

Quadratic Formula Calculator

Quadratic Formula Practice

Completing the Square

Graphing Quadratics

Standard form describes the family

Vertex form shows the turning point

Factored form shows the zeros

Use square roots when a square is isolated

Completing the square creates vertex form

The quadratic formula always works

The discriminant predicts the kind of roots

Solve 2x2−3x−2=0\displaystyle 2x^2-3x-2=02x2−3x−2=0 with the quadratic formula.

Quadratic Formula Calculator

Quadratic Formula Practice

Completing the Square

Graphing Quadratics

Solve $\displaystyle 2x^2-3x-2=0$ with the quadratic formula.