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Algebra II

The Discriminant

Lesson

The discriminant is the expression under the radical in the quadratic formula:

Δ=b2−4ac\Delta = b^2 - 4ac

Without solving the equation, the sign of Δ\Delta tells you how many real solutions a quadratic has.

  • Δ>0\Delta > 0 → two distinct real solutions.
  • Δ=0\Delta = 0 → one real solution (a double root).
  • Δ<0\Delta < 0 → no real solutions (the solutions are complex).

Worked example 1

x2+5x+6=0x^2 + 5x + 6 = 0

a=1,b=5,c=6a = 1, b = 5, c = 6.

Δ=52−4(1)(6)=25−24=1\Delta = 5^2 - 4(1)(6) = 25 - 24 = 1

Discriminant is 1 (positive) → two real solutions.

Worked example 2

x2+4x+5=0x^2 + 4x + 5 = 0

a=1,b=4,c=5a = 1, b = 4, c = 5.

Δ=16−20=−4\Delta = 16 - 20 = -4

Discriminant is −4 (negative) → no real solutions.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

x2+5x+6=0x^2 + 5x + 6 = 0

Problem 2

x2+4x+4=0x^2 + 4x + 4 = 0

Problem 3

x2+2x+5=0x^2 + 2x + 5 = 0

Problem 4

x2−6x+9=0x^2 - 6x + 9 = 0

Practice

Standard problems matching the lesson.

Problem 5

x2+7x+10=0x^2 + 7x + 10 = 0

Problem 6

x2−8x+15=0x^2 - 8x + 15 = 0

Problem 7

x2+4x+5=0x^2 + 4x + 5 = 0

Problem 8

x2−10x+25=0x^2 - 10x + 25 = 0

Problem 9

x2+3x−4=0x^2 + 3x - 4 = 0

Problem 10

x2−2x+7=0x^2 - 2x + 7 = 0

Problem 11

2x2+3x+1=02x^2 + 3x + 1 = 0

Problem 12

2x2−5x+3=02x^2 - 5x + 3 = 0

Problem 13

3x2+2x+1=03x^2 + 2x + 1 = 0

Problem 14

x2+6x+9=0x^2 + 6x + 9 = 0

Problem 15

x2−x−6=0x^2 - x - 6 = 0

Problem 16

x2+3x+5=0x^2 + 3x + 5 = 0

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

4x2−4x+1=04x^2 - 4x + 1 = 0

Problem 18

5x2−7x+1=05x^2 - 7x + 1 = 0

Problem 19

3x2+5x−2=03x^2 + 5x - 2 = 0

Problem 20

2x2+x+3=02x^2 + x + 3 = 0

Problem 21

6x2−11x+4=06x^2 - 11x + 4 = 0

Problem 22

9x2+12x+4=09x^2 + 12x + 4 = 0

Practice

Standard problems matching the lesson.

Problem 23

A profit equation: -x² + 10x − 16 = 0. Find the discriminant Δ.

Problem 24

A ball's height -5t² + 20t − 15 = 0 when it lands. Find Δ.

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

A rectangle's dimensions satisfy x² − 8x + 12 = 0. Find Δ.

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