Section 10.5 – Page 506

ObjectivesUse the quadratic formula to find solutions to quadratic equations.Use the quadratic formula to find the zeros of a quadratic function.Evaluate the discriminant to determine how many real roots a quadratic equation has and whether it can be factored.

Glossary Terms

10.5 The Quadratic Formula

discriminantquadratic formula

Rules and Properties

The Quadratic Formula

10.5 The Quadratic Formula

x = –b b2 – 4ac

2a

For ax2 + bx + c = 0, where a 0:

The Discriminant

b2 – 4ac

Use the quadratic Formula to solve x² - 10 + 3x = 0

x² + 3x – 10 = 0 Rewrite in standard form

a = _, b = _, c = _ Identify the coefficients

-(3) ± √(3)² - 4(1)(-10) 2(1)

Substitute coefficients into quadratic formula

-3 ± √ 49 2

-3 ± 7 2

-3 + 7 2

= 2

-3 – 7 2

= -5

1 3 -10

So the solutions are 2 and –5.

Find the zeroes of y = 3x² + 2x - 4

a = ___, b = ___, c = ___3 2 -4

-2 ± √2² - 4(3)(-4) 2(3)

-2 ± √4 – (-48) 6

-2 ± √52 6

-2 + √52 6

≈ 0.87

-2 - √52 6

≈ -1.54

So the solutions are 0.87 and –1.54

The Discriminant

The discriminant of a quadratic equation allows you to determine how many real-number solutions the

quadratic equation has.

b² - 4ac•If the value of the discriminant is less than 0, the equation has no real solutions (b² - 4ac < 0)•If the value of the discriminant is equal to 0, the equation has exactly one real solution (b² - 4ac = 0)•If the value of the discriminant is greater than 0, the equation has two real solutions

Find the number of real solutions to each of the following quadratic

equations

3x² - 2x + 1 = 0

a = __, b = __, c = __3 -2 1

(-2)² - 4(3)(1) ? 0

4 – 12 ? 0

-8 __ 0<

So there are no real solutions

4x = 4x² + 1

4x² - 4x + 1 = 0

a = __, b = __, c = __4 -4 1

(-4)² - 4(4)(1) ? 0

16 – 16 ? 0

0 __ 0=So there is exactly one real solution

If the discriminant is a perfect square, the equation can be factored.

Determine whether 6x² + 23x + 30 can be factored

a = 6, b = 23, c = 120

23² - 4(6)(20)

529 - 480

49

49 is a perfect square so the equation can be factored

Use the quadratic formula to find the factors

x =-23 ± √ (23)² - 4(6)(20)

2(6)

x = -23 ± √ 49 12

x = -23 + 7 12

= -16/12 = -4/3

X =-23 – 7 12

= 30/12 = -5/2

Factored Form – Cont.

x = - 4/3 x = -5/2

3x = -4

3x + 4 = 0

2x = -5

2x + 5 = 0

So the factored form is (3x + 4)(2x + 5) = 0

You can check by doing FOIL to see if you get the original equation

Assignment

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