Discriminant Examples in Math

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

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 discriminant of a quadratic equation $ax^2 + bx + c = 0$ is the expression $\Delta = b^2 - 4ac$ . It determines the number and nature of the solutions.

The discriminant is the expression under the square root in the quadratic formula. If it is positive, you can take the square root and get two answers. If it is zero, the square root is zero so both answers are the same. If it is negative, you cannot take a real square root, so there are no real solutions.

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 discriminant $b^2-4ac$ tells you the number and type of solutions before you solve.

Common stuck point: The procedure for discriminant is the easy part; the trap is computing $b^2-4ac$ with the wrong sign on $b$ . Asking "Do I only need to know how many/what kind of roots a quadratic has, not their values?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do I only need to know how many/what kind of roots a quadratic has, not their values?

Worked Examples

Example 1

easy

Find the discriminant of

x^2 - 6x + 9 = 0

and determine the number of solutions.

Answer

\Delta = 0

; one repeated solution (

x = 3

First step

Identify

a = 1, b = -6, c = 9

Full solution

2
Discriminant: $\Delta = b^2 - 4ac = 36 - 36 = 0$ .
3
Since $\Delta = 0$ , there is exactly one real solution (a repeated root).

The discriminant

\Delta = b^2 - 4ac

tells us the nature of the roots:

\Delta > 0

means two distinct real roots,

\Delta = 0

means one repeated root,

\Delta < 0

means no real roots.

Example 2

medium

For what values of

k

does

x^2 + kx + 9 = 0

have two distinct real solutions?

Example 3

medium

Use the discriminant to decide whether

3x^2 + 7x - 6 = 0

has two real solutions.

Example 4

medium

Show that

x^2 + 5x + 7 = 0

has roots that are complex conjugates with non-zero imaginary parts.

Example 5

challenge

Prove that for any real

a

, the equation

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

has two distinct real roots and find them.

Practice Problems

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

Example 1

easy

Find the discriminant of

2x^2 + 5x - 3 = 0

Example 2

medium

Without solving, how many real solutions does

3x^2 + 2x + 5 = 0

have?

Example 3

easy

Find the discriminant of

x^2 + 5x + 6 = 0

Example 4

easy

Find the discriminant of

x^2 + 4x + 4 = 0

Example 5

easy

Find the discriminant of

x^2 + x + 1 = 0

Example 6

easy

How many real solutions does an equation with

\Delta = 9

have?

Example 7

easy

How many real solutions when

\Delta = 0

Example 8

easy

How many real solutions when

\Delta = -5

Example 9

easy

Find the discriminant of

2x^2 + 3x + 1 = 0

Example 10

easy

Find the discriminant of

x^2 - 6x + 9 = 0

Example 11

medium

For what value of

k

does

x^2 + kx + 9 = 0

have exactly one real solution (

k>0

Example 12

medium

For what values of

k

does

x^2 + 4x + k = 0

have two distinct real solutions?

Example 13

medium

For what values of

k

does

x^2 + 6x + k = 0

have no real solutions?

Example 14

medium

Find the discriminant of

3x^2 - 2x - 5 = 0

Example 15

medium

Without solving, are the roots of

2x^2 + 5x - 3 = 0

real and rational?

Example 16

medium

Find the discriminant of

x^2 - 2x + 5 = 0

and state the root type.

Example 17

medium

The equation

4x^2 + 12x + 9 = 0

— how many distinct real roots?

Example 18

medium

Find the discriminant of

5x^2 - 2x + 1 = 0

Example 19

medium

For what values of

k

does

2x^2 + kx + 2 = 0

have two distinct real roots?

Example 20

challenge

For what

k

does

kx^2 + 4x + 2 = 0

have exactly one real solution? (Assume

k \ne 0

Example 21

challenge

Show that

x^2 + bx + b = 0

has no real roots for

0 < b < 4

Example 22

challenge

Find all

k

for which

x^2 + (k-1)x + 1 = 0

has a repeated root.

Example 23

easy

Find the discriminant of

x^2 + 7x + 12 = 0

Example 24

easy

Find the discriminant of

x^2 - 8x + 16 = 0

Example 25

easy

Find the discriminant of

4x^2 - 4x + 1 = 0

Example 26

easy

Find the discriminant of

x^2 + 2x + 5 = 0

Example 27

easy

Find the discriminant of

x^2 - 9 = 0

Example 28

medium

For what values of

k

does

x^2 + kx + 16 = 0

have a repeated root?

Example 29

medium

For what values of

k

does

x^2 - 2x + k = 0

have no real solutions?

Example 30

medium

For what values of

k

does

2x^2 + 3x + k = 0

have two distinct real solutions?

Example 31

medium

Without solving, classify the roots of

7x^2 - 3x + 4 = 0

Example 32

medium

Find the discriminant of

5x^2 + 10x + 5 = 0

and interpret.

Example 33

medium

Are the roots of

6x^2 - x - 1 = 0

rational?

Example 34

medium

Find the discriminant of

9x^2 + 6x + 1 = 0

and find the root.

Example 35

medium

For what values of

m

does

x^2 + mx + 4 = 0

have no real solutions?

Example 36

hard

A parabola

y = x^2 - 6x + c

is tangent to the

x

-axis. Find

c

Example 37

hard

Find all

k

for which the line

y = kx + 1

intersects

y = x^2 + 5

in exactly one point.

Example 38

hard

For what values of

p

does

x^2 + (p+1)x + p^2 = 0

have real solutions?

Example 39

hard

x^2 - 2(k+1)x + (k^2 + 4) = 0

has equal roots, find

k

Example 40

hard

Show that

x^2 + (m^2 + 1)x + m = 0

always has two distinct real roots for every real

m

Example 41

hard

The quadratic

x^2 + bx + c

has roots whose sum is

-5

and product is

4

. Use Vieta to find

b, c

, then find the discriminant.

Example 42

hard

Find the values of

k

for which

kx^2 + 4x + k = 0

has two distinct real solutions. (Assume

k \ne 0

Example 43

challenge

If both roots of

x^2 - (k+3)x + k + 2 = 0

are positive, what is the range of

k

← Back to Discriminant Formula Explained Practice Mode

Related Concepts

Quadratic Formula Quadratic Standard Form Zeros of a Quadratic Complex Numbers

Background Knowledge

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

quadratic formulaquadratic standard form