Solving Quadratic Equations

(finding roots)

Example f(x) = x2 - 4

By GraphingIdentifying Solutions

4

2

-2

-4

-5 5

Solutions are -2 and 2.

EXAMPLE 3 Solve a quadratic equation in standard form

Solve 2x2 + 20x – 8 = 0 by completing the square.

SOLUTION

Write original equation.2x2 + 20x – 8 = 0

Add 8 to each side.2x2 + 20x = 8

Divide each side by 2.x2 + 10x = 4

Add 10 2

2, or 52, to each side.x2 + 10x + 52 = 4 + 52

Write left side as the square of a binomial.

(x + 5)2 = 29

Solving Quadratic Equations by Factoring

Solve by using he zero product property.

1) 2) 3) 20 48 16t t 42 x 10133 2 xx

To solve a quadratic equation if you can’t factor the equation:

• Make sure the equation is in the general form. 

• Identify a, b, and c.

• Substitute a, b, and c into the quadratic formula:

 

• Simplify.  

• Solve a previous problem using the quadratic formula.

Descriminants can give us hints…

For the equation . . . 0742 xx

. . . the discriminant

acb 42 12

There are no real roots as the function is never equal to zero

2816

The Discriminant of a Quadratic Function

If we try to solve , we get0742 xx

2

124 x

The square of any real number is positive so there are no real solutions to 12

742 xxy0

Roots, Surds and Discriminant

Complex Conjugates and Division

Complex conjugates-a pair of complex numbers of the form a + bi and a – bi where a and b are real numbers.

( a + bi )( a – bi )

a 2 – abi + abi – b 2 i 2

a 2 – b 2( -1 )

a 2 + b 2

The product of a complex conjugate pair is a positive real number.

• Ex. Find the real and non-real roots of

154)( 2 xxxf