Quadratic Formula Examples in Math

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

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

Concept Recap

A formula giving the exact solutions to any quadratic equation $ax^2 + bx + c = 0$ directly from its three coefficients.

When factoring fails, this formula always finds the x-intercepts.

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 quadratic formula gives the exact solutions of $ax^2+bx+c=0$ from its three coefficients.

Common stuck point: The procedure for quadratic formula is the easy part; the trap is using the formula before setting the equation to zero. Asking "Do I have a quadratic equation set to zero whose exact solutions I need?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do I have a quadratic equation set to zero whose exact solutions I need?

Worked Examples

Example 1

easy

Solve

x^2 - 5x + 6 = 0

by factoring.

Answer

x = 2 \text{ or } x = 3

First step

Find two numbers that multiply to 6 and add to

-5

: those are

-2

and

-3

Full solution

2
Factor: $(x - 2)(x - 3) = 0$ .
3
Set each factor to zero: $x - 2 = 0$ or $x - 3 = 0$ .
4
Solutions: $x = 2$ or $x = 3$ .

Factoring works when you can find two numbers whose product equals the constant term and whose sum equals the coefficient of

x

. The zero product property then gives the solutions.

Example 2

medium

Solve

2x^2 + 3x - 2 = 0

using the quadratic formula.

Example 3

hard

Solve

2x^2 - 5x - 3 = 0

using the quadratic formula.

Example 4

medium

Solve

3x^2 - 4x - 4 = 0

using the quadratic formula.

Example 5

medium

A rectangular garden has area

54

m² and its length is

3

m more than its width. Find the width.

Example 6

hard

Solve

\frac{2}{x} + \frac{3}{x + 1} = 1

for

x \neq 0, -1

Practice Problems

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

Example 1

easy

Solve

x^2 - 9 = 0

Example 2

hard

Solve

x^2 + 4x + 5 = 0

and describe the solutions.

Example 3

easy

Solve

x^2 - 5x + 6 = 0

using the quadratic formula.

Example 4

easy

Solve

x^2 + 7x + 12 = 0

using the quadratic formula.

Example 5

easy

Solve

2x^2 - 4x - 6 = 0

Example 6

easy

Solve

x^2 - 9 = 0

Example 7

easy

Solve

x^2 + 2x - 8 = 0

Example 8

easy

Solve

3x^2 + 2x - 1 = 0

Example 9

easy

Solve

x^2 - 4x + 4 = 0

Example 10

easy

Find the discriminant of

5x^2 - 3x + 1 = 0

and state how many real roots the equation has.

Example 11

medium

Solve

x^2 - 2x - 1 = 0

Example 12

medium

Solve

2x^2 + 3x - 5 = 0

Example 13

medium

Solve

x^2 - 6x + 13 = 0

Example 14

medium

For what value of

k

does

x^2 - 6x + k = 0

have exactly one real solution?

Example 15

medium

A rectangle has area

24

and length

5

more than width. Find the dimensions.

Example 16

medium

A ball is thrown upward with initial velocity

20

m/s from a height of

5

m. Its height is

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

. When does it hit the ground?

Example 17

medium

Find the sum and product of the roots of

3x^2 - 7x + 2 = 0

without solving the equation.

Example 18

medium

Solve

x + \frac{6}{x} = 5

Example 19

medium

Solve

x^4 - 5x^2 + 4 = 0

Example 20

challenge

If the roots of

x^2 - 8x + k = 0

are in the ratio

3 : 1

, find

k

Example 21

challenge

Find all values of

a

for which

x^2 + ax + 4 = 0

has two distinct positive real roots.

Example 22

challenge

Prove that for any real

a

, the equation

x^2 + ax + (a - 1) = 0

has at least one real solution.

Example 23

easy

Identify

a

b

, and

c

4x^2 - 7x + 2 = 0

Example 24

easy

Solve

x^2 - 6x + 8 = 0

using the quadratic formula.

Example 25

easy

Solve

x^2 + 3x - 10 = 0

Example 26

easy

Solve

x^2 + 6x + 9 = 0

Example 27

medium

Solve

x^2 - 4x + 1 = 0

. Leave answers in exact form.

Example 28

medium

Solve

2x^2 + x - 6 = 0

Example 29

medium

Solve

x^2 - 2x + 5 = 0

in the complex numbers.

Example 30

medium

Find the values of

k

for which

x^2 + kx + 4 = 0

has a double root.

Example 31

medium

Solve

4x^2 - 12x + 9 = 0

Example 32

medium

Solve

x^2 + 5x - 14 = 0

Example 33

medium

Solve

5x^2 - 2x - 3 = 0

Example 34

hard

Solve

2x^2 + 6x + 1 = 0

. Leave answer in exact form.

Example 35

hard

A projectile follows

h(t) = -5t^2 + 25t + 1

metres. When does it reach height

20

m? Give all solutions to two decimal places.

Example 36

hard

Find values of

k

such that

x^2 + (k+2)x + 9 = 0

has no real roots.

Example 37

hard

If one root of

x^2 - 7x + k = 0

3

, find

k

and the other root.

Example 38

hard

Find

b

such that the equation

x^2 + bx + 16 = 0

has roots whose difference is

6

Example 39

hard

Solve

x^4 - 10x^2 + 9 = 0

Example 40

hard

The roots of

2x^2 - 5x + c = 0

differ by

1

. Find

c

Example 41

challenge

Prove that the equation

x^2 - (a + b)x + ab - 1 = 0

always has two distinct real roots for any real

a

b

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Related Concepts

Quadratic Functions Square Roots Complex Numbers Discriminant

Background Knowledge

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

quadratic functionssquare roots