The Discriminant

Learn how the discriminant reveals the nature of quadratic solutions before solving.

Advanced25 minLesson
Printable worksheet

Definition

The discriminant is the expression under the square root in the quadratic formula. For a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, the discriminant is:
Δ=b2−4ac\Delta = b^2 - 4ac
The discriminant tells us about the nature and number of solutions without actually solving the equation:
DiscriminantNumber of SolutionsType of Solutions
Δ>0\Delta > 0Two solutionsTwo distinct real numbers
Δ=0\Delta = 0One solutionOne repeated real number
Δ<0\Delta < 0No real solutionsTwo complex numbers
Remember the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. The discriminant is the part inside the square root!

Try it now

What is the discriminant formula for ax2+bx+c=0ax^2 + bx + c = 0?

Worked Examples

Determine the nature of solutions for x2−5x+6=0x^2 - 5x + 6 = 0
1

Identify a, b, and c

a=1a = 1, b=−5b = -5, c=6c = 6 → Coefficients identified
2

Calculate the discriminant

Δ=(−5)2−4(1)(6)=25−24\Delta = (-5)^2 - 4(1)(6) = 25 - 24 → Δ=1\Delta = 1
3

Interpret the result

Since Δ=1>0\Delta = 1 > 0, there are two distinct real solutions → Two real solutions
4

Verify by solving

x=5±12=5±12x = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2}, so x=3x = 3 or x=2x = 2 → x=2x = 2 and x=3x = 3 confirmed

Common Mistakes

Forgetting the negative sign when bb is negative

Why it's wrong: When b=−5b = -5, students calculate b2=−25b^2 = -25 instead of (−5)2=25(-5)^2 = 25

Correct: Always square the entire coefficient including its sign: (−5)2=25(-5)^2 = 25

Confusing b2−4acb^2 - 4ac with b−4acb - 4ac

Why it's wrong: Students sometimes forget to square bb, drastically changing the result

Correct: The discriminant is b2−4acb^2 - 4ac. The bb MUST be squared.

Thinking Δ=0\Delta = 0 means no solutions

Why it's wrong: Zero seems like nothing, so students assume no solutions exist

Correct: Δ=0\Delta = 0 means exactly ONE solution (a repeated root), not zero solutions

Misidentifying coefficients when equation is not in standard form

Why it's wrong: Students use wrong values for aa, bb, cc if equation is like 3x+x2=53x + x^2 = 5

Correct: Always rewrite in standard form ax2+bx+c=0ax^2 + bx + c = 0 first: x2+3x−5=0x^2 + 3x - 5 = 0

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

The Discriminant of a Quadratic Equation | Number and Type of Solutions

by Understand The Math

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Interactive Visual

Quadratic Explorer

y = x² - 6x + 9

Controls width and direction

Shifts the parabola

Y-intercept

xy

Vertex

(3, 0)

Axis of Symmetry

x = 3

Roots (x-intercepts)

x = 3 (double root)

Y-Intercept

(0, 9)

Direction

Opens upward

Discriminant

b² - 4ac = 0

Value Table

x-3-2-10123
y3625169410
VertexRootsY-InterceptAxis of Symmetry

Function graph

-2-10123456-202468xy(2, 0)(3, 0)

Tap or hover over the graph to read off values.

x² - 5x + 6 (Δ > 0)x² - 6x + 9 (Δ = 0)2x² + x + 1 (Δ < 0)
  • Zero (x-intercept): (2, 0)
  • Zero (x-intercept): (3, 0)
  • Zero (x-intercept): (3, 0)

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = x^2 - 6x + 9

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Practice Problems

15 problems
Problem 1 of 15
Easy
What is the discriminant formula for ax2+bx+c=0ax^2 + bx + c = 0?

Why It Matters

The discriminant is a powerful tool that saves time and provides insight:
  • Efficiency: Before spending time solving, check if real solutions even exist
  • Graphing: Know whether a parabola crosses the x-axis (and how many times) without graphing
  • Problem solving: In word problems, verify that your setup produces valid real-world answers
  • Engineering: Determine if a trajectory, circuit, or design has feasible solutions
Think of the discriminant as a quick diagnostic test that reveals the behavior of quadratic equations!

Real World Applications

Projectile Motion

When launching a projectile, the discriminant tells us whether it will reach a certain height.

Example:

A ball thrown upward follows h=−5t2+20t+1h = -5t^2 + 20t + 1. To find when h=25h = 25: −5t2+20t−24=0-5t^2 + 20t - 24 = 0. The discriminant Δ=400−480=−80<0\Delta = 400 - 480 = -80 < 0 tells us the ball never reaches 25 meters.

1Try It Yourself

A rocket's height is modeled by h=−4t2+32th = -4t^2 + 32t. You want to know if it reaches 64 meters.

Does the rocket reach 64 meters? Use the discriminant to decide.

Step 1: Write the mathematical expression

Set up −4t2+32t=64-4t^2 + 32t = 64, then calculate Δ\Delta:

Business Break-Even Analysis

Companies use the discriminant to determine if profit targets are achievable.

Example:

If profit P=−2x2+40x−100P = -2x^2 + 40x - 100, finding when P=150P = 150 gives −2x2+40x−250=0-2x^2 + 40x - 250 = 0. With Δ=1600−2000=−400<0\Delta = 1600 - 2000 = -400 < 0, profit of 150 is impossible.

2Try It Yourself

A company's daily profit is P=−x2+12x−20P = -x^2 + 12x - 20 thousand euros, where xx is items sold (in hundreds).

Can they achieve a profit of 16 thousand euros?

Step 1: Write the mathematical expression

Set P=16P = 16: −x2+12x−36=0-x^2 + 12x - 36 = 0. Find Δ\Delta:

Key Takeaways

  • 1The discriminant is Δ=b2−4ac\Delta = b^2 - 4ac for the equation ax2+bx+c=0ax^2 + bx + c = 0
  • 2If Δ>0\Delta > 0: two distinct real solutions
  • 3If Δ=0\Delta = 0: one repeated real solution (double root)
  • 4If Δ<0\Delta < 0: no real solutions (two complex solutions)
  • 5The discriminant appears under the square root in the quadratic formula
  • 6Use the discriminant to predict solutions without fully solving the equation

Frequently Asked Questions

The word discriminant comes from discriminate meaning to distinguish or tell apart. The discriminant discriminates between different types of solutions: two real, one real, or no real solutions.
The word discriminant comes from discriminate meaning to distinguish or tell apart. The discriminant discriminates between different types of solutions: two real, one real, or no real solutions.
The solutions become complex numbers involving i=−1i = \sqrt{-1}. For example, if Δ=−4\Delta = -4, then Δ=−4=2i\sqrt{\Delta} = \sqrt{-4} = 2i. Complex solutions are beyond most algebra courses but are studied in advanced math.
Absolutely! The discriminant tells you how many x-intercepts the parabola has: Δ>0\Delta > 0 means 2 x-intercepts, Δ=0\Delta = 0 means 1 x-intercept (vertex touches x-axis), Δ<0\Delta < 0 means no x-intercepts.

Glossary

Discriminant
The expression b2−4acb^2 - 4ac that determines the nature of solutions to a quadratic equation
Double root
A repeated solution that occurs when the discriminant equals zero
Real solution
A solution that is a real number (not involving ii)
Complex solution
A solution involving imaginary numbers, occurring when Δ<0\Delta < 0

Formula Card

Discriminant

Δ=b2−4ac\Delta = b^2 - 4ac

The expression that determines the nature of solutions

Quadratic Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Formula for solving any quadratic equation

Two Solutions

Δ>0\Delta > 0

Condition for two distinct real solutions

One Solution

Δ=0\Delta = 0

Condition for one repeated real solution

No Real Solutions

Δ<0\Delta < 0

Condition for no real solutions

Today's challengeOne problem, three tries

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