4. The Discriminant

Learning outcomes

I can calculate the discriminant.
I can predict the number of real solutions.
I can distinguish between repeated and distinct roots.
I can explain what the discriminant represents graphically.
I can relate the discriminant to x-intercepts.

The Discriminant

What Is the Discriminant?

A quadratic equation can be written in standard form:

ax² + bx + c = 0

The discriminant is the expression:

D = b² − 4ac

The discriminant is part of the quadratic formula:

x = [−b ± √(b² − 4ac)] / (2a)

The value of the discriminant tells us important information about the solutions before we actually solve the quadratic.

It can tell us whether the equation has:

two distinct real solutions
one repeated real solution
no real solutions

These solutions correspond directly to the x-intercepts of the parabola.

Identifying a, b and c

Before calculating the discriminant, write the quadratic in standard form:

ax² + bx + c = 0

Then identify:

a = coefficient of x²
b = coefficient of x
c = constant

For example:

2x² − 7x + 3 = 0

gives:

a = 2

b = −7

c = 3

The negative sign is part of the coefficient.

Calculating the Discriminant

For:

2x² − 7x + 3 = 0

use:

D = b² − 4ac

Substitute:

D = (−7)² − 4(2)(3)

D = 49 − 24

D = 25

Because the discriminant is positive, this quadratic has two distinct real solutions.

We did not need to solve the equation to determine this.

The Three Possible Cases

The sign of the discriminant determines the number of real roots.

Discriminant	Roots	x-Intercepts
D > 0	Two distinct real roots	Two
D = 0	One repeated real root	One
D < 0	No real roots	None

This relationship is one of the most useful features of the discriminant.

Positive Discriminant

If:

D > 0

then the quadratic has:

two distinct real roots

The word distinct means that the roots have different values.

For example:

x² − 5x + 6 = 0

Identify:

a = 1

b = −5

c = 6

Calculate:

D = (−5)² − 4(1)(6)

D = 25 − 24

D = 1

Since:

D > 0

there are two distinct real roots.

In fact:

x² − 5x + 6 = (x − 2)(x − 3)

so the roots are:

x = 2 and x = 3

Positive Discriminant and the Graph

For the function:

y = x² − 5x + 6

the graph crosses the x-axis at:

(2, 0)

and:

(3, 0)

So a positive discriminant means that the parabola has two different x-intercepts.

The roots are the x-values of those intercepts.

Zero Discriminant

If:

D = 0

the quadratic has:

one repeated real root

Consider:

x² − 6x + 9 = 0

Here:

a = 1

b = −6

c = 9

Calculate:

D = (−6)² − 4(1)(9)

D = 36 − 36

D = 0

Therefore, the equation has one repeated root.

Factor:

(x − 3)² = 0

So:

x = 3

What Is a Repeated Root?

A repeated root is sometimes called a:

double root
repeated solution
equal root

For:

(x − 3)² = 0

we could write:

(x − 3)(x − 3) = 0

Both factors produce the same solution:

x = 3

Therefore, there is only one distinct real solution, even though the root occurs twice algebraically.

Zero Discriminant and the Graph

When:

D = 0

the parabola touches the x-axis at exactly one point.

That point is also the vertex.

For:

y = (x − 3)²

the vertex is:

(3, 0)

The graph touches the x-axis but does not pass through it.

This is why a repeated root corresponds to exactly one x-intercept.

Negative Discriminant

If:

D < 0

there are:

no real roots

Consider:

x² + 2x + 5 = 0

Here:

a = 1

b = 2

c = 5

Calculate:

D = 2² − 4(1)(5)

D = 4 − 20

D = −16

Since:

D < 0

the equation has:

no real solutions

Why a Negative Discriminant Gives No Real Roots

The quadratic formula contains:

√D

If D is negative, we would need to calculate the square root of a negative number.

For example:

√(−16)

There is no real number whose square is −16.

Therefore, there are no real solutions.

If complex numbers are being used, the equation can still be solved using the imaginary unit i.

Negative Discriminant and the Graph

If a quadratic has no real roots, its graph never reaches the x-axis.

For example:

y = x² + 2x + 5

can be rewritten:

y = (x + 1)² + 4

Its vertex is:

(−1, 4)

Since the parabola opens upward and its minimum y-value is 4, it remains completely above the x-axis.

Therefore:

no real roots
no x-intercepts
negative discriminant

These are three ways of describing the same situation.

Roots and x-Intercepts

For a quadratic function:

y = ax² + bx + c

the x-intercepts occur when:

y = 0

Therefore, we solve:

ax² + bx + c = 0

The solutions are the roots.

So:

roots ↔ x-intercepts

This explains why the discriminant tells us about both algebraic solutions and graphical intersections.

Comparing the Three Cases

Imagine three upward-opening parabolas.

Two x-intercepts

The parabola crosses the x-axis twice.

D > 0

There are two distinct real roots.

One x-intercept

The vertex lies exactly on the x-axis.

D = 0

There is one repeated real root.

No x-intercepts

The entire parabola remains above the x-axis.

D < 0

There are no real roots.

The same relationships also apply to downward-opening parabolas.

Distinct Roots

Distinct roots simply means that the solutions have different values.

For example:

x = −2 and x = 5

are distinct roots.

A quadratic with distinct real roots has:

D > 0

and its graph has:

two different x-intercepts

Repeated Roots

A repeated root occurs when both solutions have the same value.

For example:

x = 4

might occur twice:

(x − 4)(x − 4) = 0

or:

(x − 4)² = 0

This occurs when:

D = 0

Graphically, the vertex lies exactly on the x-axis.

Example: Predict Without Solving

Consider:

3x² + 4x − 2 = 0

Calculate:

D = 4² − 4(3)(−2)

D = 16 + 24

D = 40

Since:

40 > 0

we know immediately:

two distinct real roots
two x-intercepts

We do not need to calculate the actual roots unless the question asks for them.

Another Prediction

Consider:

4x² − 12x + 9 = 0

Calculate:

D = (−12)² − 4(4)(9)

D = 144 − 144

D = 0

Therefore:

one repeated real root
one x-intercept
the vertex lies on the x-axis

Another Prediction

Consider:

2x² + 3x + 7 = 0

Calculate:

D = 3² − 4(2)(7)

D = 9 − 56

D = −47

Therefore:

no real roots
no x-intercepts
the parabola does not cross or touch the x-axis

Perfect-Square Discriminants

A positive discriminant tells us there are two real roots, but it can tell us even more.

Suppose:

D = 25

Then:

√D = 5

Since 25 is a perfect square, the quadratic formula will usually produce rational roots.

For example:

2x² − 7x + 3 = 0

has:

D = 25

and roots:

x = 3 and x = 1/2

Both are rational.

Positive but Not a Perfect Square

Suppose:

D = 20

Then:

√20 = 2√5

The roots are real, but they usually contain radicals.

For example:

x² − 4x − 1 = 0

has:

D = (−4)² − 4(1)(−1)

D = 16 + 4

D = 20

Therefore, there are two distinct real roots.

Using the quadratic formula gives:

x = 2 ± √5

These roots are irrational.

What the Size of the Discriminant Does Not Tell Us Directly

It is tempting to think that a larger discriminant always means the graph has x-intercepts that are farther apart.

The relationship is more precise.

From the quadratic formula:

x₁ = [−b + √D]/(2a)

x₂ = [−b − √D]/(2a)

The distance between the roots is:

|x₁ − x₂| = √D/|a|

So the separation depends on both:

the discriminant
the coefficient a

The sign of D determines the number and type of real roots, while other coefficients also affect their locations.

The Discriminant and the Vertex

The discriminant can also tell us how the vertex relates to the x-axis.

For:

y = ax² + bx + c

the vertex occurs at:

x = −b/(2a)

Its y-coordinate can be written as:

y_vertex = −D/(4a)

This creates an interesting connection between the discriminant and the graph.

If the Parabola Opens Upward

When:

a > 0

the parabola opens upward.

If D > 0:

y_vertex < 0

so the vertex lies below the x-axis and the parabola crosses the axis twice.

If D = 0:

y_vertex = 0

so the vertex lies on the x-axis.

If D < 0:

y_vertex > 0

so the vertex lies above the x-axis and there are no x-intercepts.

If the Parabola Opens Downward

When:

a < 0

the parabola opens downward.

The same discriminant rules still apply:

D > 0 → two x-intercepts

D = 0 → one x-intercept

D < 0 → no x-intercepts

The parabola's orientation changes, but the discriminant rules do not.

The Discriminant in Factored Form

Suppose a quadratic can be written:

y = a(x − r₁)(x − r₂)

If:

r₁ ≠ r₂

there are two distinct roots.

Therefore:

D > 0

If:

r₁ = r₂

there is a repeated root.

Therefore:

D = 0

If there are no real values for r₁ and r₂, then:

D < 0

So the discriminant connects standard form with what we see in factored form.

The Discriminant and the Quadratic Formula

The quadratic formula is:

x = [−b ± √D]/(2a)

This makes the importance of the discriminant clear.

If D > 0

√D is a positive real number.

The + and − produce different answers.

Two distinct real roots.

If D = 0

√D = 0.

The + and − produce the same answer.

One repeated real root.

If D < 0

√D is not a real number.

No real roots.

This is exactly why the discriminant works.

Worked Example: Two Distinct Roots

Determine the nature of the roots of:

2x² − 5x − 3 = 0

Identify:

a = 2

b = −5

c = −3

Calculate:

D = (−5)² − 4(2)(−3)

D = 25 + 24

D = 49

Since:

D > 0

there are:

two distinct real roots

Since 49 is also a perfect square, the roots will be rational.

Worked Example: Repeated Root

Determine the nature of the roots of:

9x² + 12x + 4 = 0

Identify:

a = 9

b = 12

c = 4

Calculate:

D = 12² − 4(9)(4)

D = 144 − 144

D = 0

Therefore:

one repeated real root

In fact:

9x² + 12x + 4 = (3x + 2)²

So:

x = −2/3

Worked Example: No Real Roots

Determine the nature of the roots of:

3x² − 2x + 5 = 0

Identify:

a = 3

b = −2

c = 5

Calculate:

D = (−2)² − 4(3)(5)

D = 4 − 60

D = −56

Therefore:

no real roots

and:

no x-intercepts

Worked Example: Missing Term

Consider:

x² − 16 = 0

There is no x-term, so:

b = 0

Identify:

a = 1

b = 0

c = −16

Calculate:

D = 0² − 4(1)(−16)

D = 64

Since:

D > 0

there are two distinct real roots.

In fact:

x = ±4

Rearrange Before Calculating

Consider:

x² + 6 = 5x

Do not calculate the discriminant yet.

First rearrange:

x² − 5x + 6 = 0

Now:

a = 1

b = −5

c = 6

Calculate:

D = 25 − 24

D = 1

Therefore, there are two distinct real roots.

Using the Discriminant to Find Unknown Values

The discriminant can also help determine when a quadratic has a certain number of solutions.

Consider:

x² + kx + 9 = 0

Suppose we want the equation to have one repeated real root.

For a repeated root:

D = 0

So:

k² − 4(1)(9) = 0

k² − 36 = 0

k² = 36

Therefore:

k = ±6

So either:

x² + 6x + 9 = 0

or:

x² − 6x + 9 = 0

has a repeated root.

Discriminant Conditions

For:

ax² + bx + c = 0

we can use inequalities.

For two distinct real roots:

b² − 4ac > 0

For one repeated real root:

b² − 4ac = 0

For no real roots:

b² − 4ac < 0

This makes the discriminant useful not only for solving equations but also for analyzing entire families of quadratic equations.

Graphical Interpretation

The discriminant answers a very specific graphical question:

How many times does the parabola meet the x-axis?

If:

D > 0

the parabola crosses the x-axis at two different points.

If:

D = 0

the parabola touches the x-axis at its vertex.

If:

D < 0

the parabola never reaches the x-axis.

This connection makes the discriminant both an algebraic tool and a graphical tool.

Why the Discriminant Is Useful

Without the discriminant, we might have to:

factor the quadratic
complete the square
use the entire quadratic formula
graph the function

just to determine how many real roots exist.

The discriminant lets us answer that question quickly.

For example:

7x² − 3x + 12 = 0

D = (−3)² − 4(7)(12)

D = 9 − 336

D = −327

We immediately know:

no real solutions

There is no need to continue solving if the question only asks about the number of real roots.

Common Mistakes

Mistake: Using D = b² + 4ac.
The correct formula is:

D = b² − 4ac

Mistake: Losing the sign of b.
If b = −5, calculate:

(−5)² = 25

Mistake: Forgetting that c can be negative.
For c = −3:

−4ac may become positive.

Mistake: Calculating the discriminant before rearranging into standard form.
First write:

ax² + bx + c = 0

Mistake: Saying D > 0 means exactly two solutions of every type.
It means two distinct real solutions.

Mistake: Saying D = 0 means no solutions.
It means one repeated real root.

Mistake: Saying D < 0 means the equation has no solutions at all.
It has no real solutions. Complex solutions may exist.

Mistake: Confusing roots with points.
A root might be x = 3, while the corresponding x-intercept is (3, 0).

Key Terms

Discriminant — expression D = b² − 4ac that determines the nature of the roots of a quadratic.

Quadratic equation — equation that can be written as ax² + bx + c = 0.

Coefficient — number multiplying a variable or power of a variable.

Root — x-value that makes a quadratic equal to zero.

Solution — value that satisfies an equation.

Real root — root belonging to the real number system.

Distinct roots — roots with different values.

Repeated root — root occurring twice in a quadratic equation.

Double root — another name for a repeated root.

x-intercept — point where a graph meets the x-axis.

Perfect square — number such as 1, 4, 9, 16, 25 or 36 that is the square of an integer.

Irrational root — real root that cannot be written as a ratio of integers.

Quadratic formula — general formula for finding the roots of a quadratic equation.

Key Takeaways

The discriminant is D = b² − 4ac.
Calculate it using the coefficients from ax² + bx + c = 0.
Always rearrange the quadratic into standard form first.
If D > 0, there are two distinct real roots.
If D = 0, there is one repeated real root.
If D < 0, there are no real roots.
Roots correspond to the x-intercepts of a quadratic graph.
A positive discriminant means the parabola has two x-intercepts.
A zero discriminant means the parabola touches the x-axis at one point.
A negative discriminant means the parabola has no x-intercepts.
A repeated root occurs at the vertex of a parabola.
If D is a positive perfect square, the roots are typically rational.
If D is positive but not a perfect square, the roots are typically irrational.
The discriminant allows us to predict the nature of the roots without fully solving the equation.
The discriminant appears under the square-root sign in the quadratic formula.
Its sign determines whether that square root produces two real values, zero, or no real value.
The discriminant connects algebraic solutions directly with the graph of a quadratic.

Check Your Understanding

State the formula for the discriminant.
Calculate the discriminant of x² − 7x + 10 = 0.
Calculate the discriminant of x² − 8x + 16 = 0.
Calculate the discriminant of x² + 4x + 10 = 0.
Determine the number of real roots of 2x² − 5x − 3 = 0.
Determine the number of real roots of 3x² + 6x + 3 = 0.
Determine the number of real roots of 4x² + 2x + 7 = 0.
Explain what D > 0 tells you about the graph.
Explain what D = 0 tells you about the graph.
Explain what D < 0 tells you about the graph.
What is the difference between a distinct root and a repeated root?
Explain the relationship between roots and x-intercepts.
A quadratic has D = 36. How many real roots does it have? What else can you predict about the roots?
A quadratic has D = 7. What can you predict about its roots?
A quadratic has D = −12. What can you predict about its graph?
Determine the discriminant of x² + 8 = 6x after first writing the equation in standard form.
Find the value or values of k that make x² + kx + 16 = 0 have one repeated real root.
Explain why a negative discriminant produces no real x-intercepts.
Explain why the ± part of the quadratic formula produces only one distinct root when D = 0.
Describe how you could use the discriminant to predict the appearance of a parabola before graphing it.