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9.6. Applications of the Discriminant. Objectives: To use and apply the discriminant to find the number of solutions to a quadratic equation. The discriminant is the expression inside the radical in the quadratic formula, b 2 – 4 ac.

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9.6

Applications of the Discriminant

Objectives: To use and apply the discriminant to find the number of solutions to a quadratic equation.

The discriminant is the expression inside the radical in the quadratic formula, b2 – 4ac.

• If b2 – 4ac is positive, then the equation has two solutions.

• If b2 – 4acis zero, then the equation has one solution.

• Ifb2 – 4acis negative, then the equation has no real solution.

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The discriminant also tells the number of times the parabola crosses the x-axis

Positive discriminant: The parabola crosses x-axis twice.

Zero discriminant: The parabola crosses x-axis once.

Negative discriminant: The parabola never crosses x-axis.

Positive

Two solutions

Negative

No solutions

Zero

One solution

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Examples: Find the discriminant and determine the number of solutions

a =1

b =-3

c =-4

1.

9 - -16

25 Two solutions

a =1

b =2

c =5

2.

4 – 20

–16 No solutions

3.

16 – 16

0 One solution