4.6 The Quadratic Formula and the Discriminant Objectives: 1.Solve quadratic equations by using the...

4.6 The Quadratic Formula and the Discriminant

Objectives:1. Solve quadratic equations by using

the Quadratic Formula2. Use the discriminant to determine

the number and type of roots for a quadratic equation.

Quadratic Formula

Always works to solve a quadratic equation, but is a little lengthy.

The solutions of a quadratic equation of the form where a≠0 are given by the formula:

2 4

2

b b acx

a

2 0ax bx c

Example

x²-8x=33x²-8x-33=0 Set = 0a=1, b=-8, c=-33

2 4

2

b b acx

a

28 ( 8) 4(1)( 33)

2(1)x

8 64 132

2x

8 196

2x

8 14

2x

8 14

2x

8 14

2x

22

2x 6

2x

11x 3x { 3,11}

Another Example7x²+6x+2=0a=7, b=6, c=2

Since ALL of the coefficients are divisible by 2, simplify by dividing them by 2.

26 (6) 4(7)(2)

2(7)x

6 36 56

14x

6 20

14x

6 2 5

14

ix

3 5

7

ix

Discriminant

The discriminant describes the solution to a quadratic equation. The part of the quadratic formula under the radical is the discriminant or b²-4ac.

• If b2 – 4ac > 0, and a perfect square– You have two rational roots

• If b2 – 4ac >0, and not a perfect square.– You have two irrational roots

• If b2 – 4ac = 0– You have 1 real, rational root. (Repeated root)

• If b2 – 4ac < 0– You have two complex roots

ExamplesFind the discriminant and

describe the number and type of roots.

a. x²-16x+64=0b²-4ac

(-16)²-4(1)(64)=256-256=0

One real, rational root because the discriminant equals 0.

b. 7x²-3x=0(-3)²-4(7)(0)=9-0=9

Two rational roots because 9 is positive and a perfect square.

c. 3x²-x+5=0(-1)²-4(3)(5)=1-60=-59

Two complex roots because the discriminant is a negative.

We have discussed several methods for solving quadratic equations – which one do you use?

Method Can Be Used When to Use

Graphing sometimes Use only if an exact number is not required. Best use to check the reasonableness of solutions found algebraically

Factoring sometimes Use if the constant term is 0 or if the factors are easily determined

Square Root Property sometimes Use for equations in which a perfect square is equal to a constant

Completing the Square always Useful for equations of the form ax2 + bx + c where b is even

Quadratic Formula always When other methods fail or are too tedious

Solve – use any method.1. 7x²-14x=07x(x-2)=0x=0, x=22. x²-64=0

x²=64x=8

3. x²-16x+64=0(x-8)(x-8)=0x=8

4. x²+5x+8=0Doesn’t factor, not easily

done by completing the square (5 is odd) so use quadratic formula.

25 5 4(1)(8)

2(1)x

5 25 32

2x

5 7

2x

5 7

2

ix

HomeworkWorkbook Page 55