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2-2: Solving Quadratic Equations Algebraically

© 2007 Roy L. Gover (www.mrgover.com)

Learning Goals:•Solve by factoring•Solve by taking square root of both sides•Solve by completing the square•Solve by using quadratic formula

DefinitionA quadratic, or second degree equation is one that can be written in the form2 0ax bx c for real constants a,b, and c with a≠0. This is the standard form for a quadratic equation.

Important IdeaThere are 4 techniques to algebraically solve quadratic equations:•Factoring•Taking square root of both sides•Completing the square•Using quadratic formula

ExampleSolve by factoring:

23 10x x

Definition

The zero product property: If the product of real numbers is zero, then one or both of the numbers must be zero

Example

Solve by factoring:

What is wrong with this:

2 6

( 1) 6

6 & 1 6 7

x x

x x

x x x

Try ThisSolve by factoring:

22 3 1 0t t 1

& 12

t t

Can you think of a way to check your answer?

Try ThisSolve by factoring:

218 23 6x x

2 3 or

9 2x x

Hint: write in standard form

ExampleSolve by taking the square root of both sides:

24 16x a.

22 15x b.

Try ThisSolve by taking the square root of both sides:

24 16x

2x

Try ThisSolve by taking the square root of both sides. Give exact and approximate solutions.23 16x

4 3 2.309

3x

Example

Solve by taking the square root of both sides:

What is wrong with this?

2 4x

ExampleComplete the square for:

2 12x x1. Half the coefficient of x: 1/2 of 12=6

2. Square this number and add to the expression

36

Important Idea

Completing the square is the process of finding the number that will make the expression a perfect square trinomial.

Important Idea2 12 36x x

is a perfect square trinomial because it factors as:

2( 6)( 6) ( 6)x x x

Try This

Complete the square for: 2 8x xthen factor your result

2 28 16 ( 4)x x x

ExampleComplete the square for: 2 3

4y y

then factor your result.Use fractions only.

ExampleSolve by completing the square: 2 8 14 0x x 1. Move the constant to the right:

2 8 14x x

ExampleSolve by completing the square: 2 8 14 0x x

2. Complete the square and add to the left and right:

2 8 16 14 16x x

Example

2( 4) 2x

4 2x

Solve by completing the square: 2 8 14 0x x 3. Factor left side and solve:

4 2x

Try ThisSolve by completing the square:

2 4 1 0x x

2 3x

ExampleSolve by completing the square…

26 2 0x x Before you complete the square, the coefficient of the squared term must be 1

Try ThisSolve by completing the square…fractions only.

22 13 15 0x x 3

, 52

x

DefinitionThe solutions to

2 0ax bx c

are:2 4

2

b b acx

a

These solutions are called the Quadratic Formula

Important Idea2 4

2

b b acx

a

2 4b ac is called thediscriminant

1. If 2 4 0b ac there are 2 real solutions

Important Idea2 4

2

b b acx

a

2 4b ac is called thediscriminant

2. If 2 4 0b ac there is 1 real solution

Important Idea2 4

2

b b acx

a

2 4b ac is called thediscriminant

3. If 2 4 0b ac there are no real solutions

ExampleSolve using the quadratic formula. Leave answer in simplified radical form.

2 2 1x x

Try ThisSolve using the quadratic formula. Leave answer in simplified radical form.24 3 5x x

3 89

8x

ExampleSolve using the quadratic formula. Leave answer in simplified radical form.4 24 13 3 0x x

Lesson Close

State the quadratic formula from memory.