Zeros of a Quadratic Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Zeros of a Quadratic.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The zeros (or roots) of a quadratic function $f(x) = ax^2 + bx + c$ are the values of $x$ where $f(x) = 0$ . Graphically, they are the $x$ -intercepts of the parabola.

The zeros are where the parabola crosses or touches the $x$ -axis. A parabola can cross twice (two zeros), just touch once (one repeated zero), or miss entirely (no real zeros). You can find them by factoring, completing the square, or using the quadratic formula.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The zeros are the x-values where a quadratic equals zero — its x-intercepts.

Common stuck point: The procedure for zeros of a quadratic is the easy part; the trap is taking only the $+$ branch of $\pm$ . Asking "Am I looking for the x-values that make the quadratic equal zero?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I looking for the x-values that make the quadratic equal zero?

Worked Examples

Example 1

easy

Find the zeros of

f(x) = x^2 - 7x + 10

Answer

x = 2 \text{ and } x = 5

First step

Set

f(x) = 0

x^2 - 7x + 10 = 0

Full solution

2
Factor: $(x - 2)(x - 5) = 0$ .
3
Zeros: $x = 2$ and $x = 5$ .

The zeros of a function are the

x

-values where

f(x) = 0

. They correspond to the

x

-intercepts of the graph.

Example 2

medium

Find the zeros of

g(x) = 2x^2 - 8x

Example 3

medium

Find the zeros of

f(x) = x^2 + 6x + 5

by factoring.

Example 4

hard

A ball's height (m) is

h(t) = -5t^2 + 20t

. When does it hit the ground?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the zeros of

h(x) = x^2 - 1

Example 2

hard

How many zeros does

f(x) = x^2 + 4

have over the reals?

Example 3

easy

Find the zeros of

f(x) = (x-4)(x+2)

Example 4

easy

Find the zeros of

f(x) = x^2 - 9

Example 5

easy

How many real zeros does

f(x) = x^2 + 1

have?

Example 6

easy

Find the zeros of

f(x) = x^2 - 5x + 6

Example 7

easy

Find the zero of

f(x) = (x-7)^2

Example 8

easy

f(0)

a zero of

f(x) = x^2 - 4

Example 9

easy

Find the zeros of

f(x) = 2x^2 - 8

Example 10

easy

How many zeros does

f(x) = x^2 - 6x + 9

have?

Example 11

medium

Find the zeros of

f(x) = x^2 - 2x - 8

by factoring.

Example 12

medium

Find the zeros of

f(x) = x^2 - 4x + 1

using the quadratic formula.

Example 13

medium

Find the zeros of

f(x) = 2x^2 + 5x + 2

Example 14

medium

Use the discriminant to find how many real zeros

f(x)=x^2+3x+5

has.

Example 15

medium

3

is a zero of

f(x) = x^2 + bx - 12

, find

b

Example 16

medium

Find the zeros of

f(x) = x^2 + 4x

and the

y

-intercept.

Example 17

medium

The zeros of a quadratic are

-2

and

6

. Find the axis of symmetry.

Example 18

medium

Find the zeros of

f(x) = x^2 + 7x + 10

Example 19

medium

-4

is a zero of

f(x) = x^2 + bx + 4

, find

b

Example 20

challenge

A quadratic

f(x)=x^2+bx+c

has zeros

2

and

-5

. Find

b

and

c

Example 21

challenge

Find the zeros of

f(x) = x^2 - 6x + 7

and simplify the radical.

Example 22

challenge

For what

c

does

f(x) = x^2 - 8x + c

have a repeated zero, and what is it?

Example 23

easy

Find the zeros of

f(x) = (x - 3)(x + 8)

Example 24

easy

Find the zeros of

f(x) = x^2 - 16

Example 25

easy

Find the zeros of

f(x) = x^2 + x - 6

Example 26

easy

Find the zeros of

f(x) = 3x^2 - 12x

Example 27

easy

Find the zeros of

f(x) = x^2 - 100

Example 28

medium

Find the zeros of

f(x) = x^2 - 10x + 25

Example 29

medium

Use the quadratic formula to find the zeros of

f(x) = x^2 + 2x - 4

Example 30

medium

Find the zeros of

f(x) = 3x^2 - 7x + 2

Example 31

medium

5

is a zero of

f(x) = x^2 - 8x + c

, find

c

Example 32

medium

The zeros of a quadratic are

-3

and

7

. Find the axis of symmetry.

Example 33

medium

Find the sum and product of the zeros of

f(x) = 2x^2 + 5x - 3

Example 34

medium

Find the zeros of

f(x) = x^2 - 6x + 9

Example 35

medium

Find the zeros of

f(x) = -x^2 + 5x + 6

Example 36

hard

Find the zeros of

f(x) = x^2 - 10x + 18

(use the quadratic formula and simplify).

Example 37

hard

If a quadratic with leading coefficient

1

has zeros

-1

and

4

, write it in standard form.

Example 38

hard

For what value(s) of

k

does

f(x) = x^2 + kx + 9

have a repeated zero?

Example 39

hard

For what value(s) of

k

does

f(x) = x^2 + kx + 4

have no real zeros?

Example 40

hard

Find the zeros of

f(x) = 4x^2 - 4x - 3

Example 41

hard

Find the zeros of

f(x) = x^2 + 4x + 1

by completing the square.

Example 42

challenge

A quadratic

f(x) = ax^2 + bx + c

has zeros

r

and

s

with

r + s = 8

and

rs = 12

. Find

b/a

and

c/a

Example 43

challenge

f(x) = x^2 + px + q

has zeros

r

and

s

with

r + s = 5

and

r^2 + s^2 = 13

, find

p

and

q

← Back to Zeros of a Quadratic Formula Explained Practice Mode

Related Concepts

Quadratic Functions Factoring Quadratic Formula Discriminant Quadratic Factored Form Polynomials

Background Knowledge

These ideas may be useful before you work through the harder examples.

quadratic functionsfactoringquadratic formula