7.4 The Quadratic Formula and the Discriminant

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7.4 The Quadratic Formula and the Discriminant. Algebra 2 Mrs. Spitz Spring 2007. Objectives. Solve equations using the quadratic formula. Use the discriminant to determine the nature of the roots of a quadratic equation. Assignment. pp. 330-331 #7-33 odd.

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### 7.4 The Quadratic Formula and the Discriminant

Algebra 2

Mrs. Spitz

Spring 2007

Objectives
• Solve equations using the quadratic formula.
• Use the discriminant to determine the nature of the roots of a quadratic equation.
Assignment
• pp. 330-331 #7-33 odd
Derive the quadratic formula from ax2 + bx + c = 0 a≠ 0

General form of a quadratic equation.

Divide all by a

Simplify

Subtract c/a on both sides.

Multiply by ½ and square the result.

Derive the quadratic formula from ax2 + bx + c = 0 a≠ 0

Add the result to both sides.

Simplify

Multiply by common denominator

Simplify

Derive the quadratic formula from ax2 + bx + c = 0 a≠ 0

Square root both sides

Simplify

Common denominator/subtract from both sides

Simplify

• The solutions of a quadratic equation of the form ax2 + bx + c with a ≠ 0 are given by this formula:

MEMORIZE!!!!

Ex. 1: Solve t2 – 3t – 28 = 0

a = 1 b = -3 c = -28

There are 2 distinct roots—Real and rational.

CHECK:

t2 – 3t – 28 = 0

72 – 3(7) – 28 = 0

49 – 21 – 28 = 0

49 – 49 = 0 

CHECK:

t2 – 3t – 28 = 0

(-4)2 – 3(-4) – 28 = 0

16 + 12 – 28 = 0

28 – 28 = 0 

Ex. 1: Solve t2 – 3t – 28 = 0
Ex. 2: Solve x2 – 8x + 16 = 0

a = 1 b = -8 c = 16

There is 1 distinct root—Real and rational.

Ex. 2: Solve x2 – 8x + 16 = 0

CHECK:

x2 – 8x + 16 = 0

(4)2 – 8(4) + 16 = 0

16 – 32 + 16 = 0

32 – 32 = 0 

There is 1 distinct root—Real and rational.

Ex. 3: Solve 3p2 – 5p + 9 = 0

a = 3 b = -5 c = 9

There is 2 imaginary roots.

Ex. 3: Solve 3p2 – 5p + 9 = 0

NOTICE THAT THE PARABOLA DOES NOT TOUCH THE X-AXIS.

Note:
• These three examples demonstrate a pattern that is useful in determining the nature of the root of a quadratic equation. In the quadratic formula, the expression under the radical sign, b2 – 4ac is called the discriminant. The discriminant tells the nature of the roots of a quadratic equation.
DISCRIMINANT
• The discriminant will tell you about the nature of the roots of a quadratic equation.
Ex. 4: Find the value of the discriminant of each equation and then describe the nature of its roots.

2x2 + x – 3 = 0

a = 2 b = 1 c = -3

b2 – 4ac = (1)2 – 4(2)(-3)

= 1 + 24

= 25

The value of the discriminant is positive and a perfect square, so 2x2 + x – 3 = 0 has two real roots and they are rational.

Ex. 5: Find the value of the discriminant of each equation and then describe the nature of its roots.

x2 + 8 = 0

a = 1 b = 0 c = 8

b2 – 4ac = (0)2 – 4(1)(8)

= 0 – 32

= – 32

The value of the discriminant is negative, so x2 + 8 = 0 has two imaginary roots.