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x2 – 7x – 18 = 0 → This problem can be solved using the quadratic formula, but it would be faster to solve the problem by factoring Generally, factoring should be considered if the leading coefficient is a 1, meaning the problem is in the form x2 + bx + c = 0

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Example 7: Solve using the quadratic formula, express your answer as an exact value 2 − 3 = 14 2 

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1.4 -11.1 -1

Quadratic Equations

1.4

Solving a Quadratic Equation

Completing the Square

The Quadratic Formula

1.4 -21.4 -2

Quadratic Equation in One

Variable

An equation that can be written in the

form 2

0ax bx c

where a, b, and care real numbers with aquadratic equation.

The given form is called standard

form.

1.4 -31.4 -3

Second

-degree Equation

A quadratic equation is a second-degree

equation.

This is an equation with a squared variable

term and no terms of greater degree. 222

25, 4 4 5 0, 3 4 8

1.4 -41.4 -4

Zero-Factor Property

If aand bare complex numbers with

ab= 0, then a= 0 or b= 0 or both.

1.4 -51.4 -5

Example 1USING THE ZERO-FACTOR

PROPERTY

Solve

Solution:

2

6 73xx

2

6 73xx

2

6 7 30XX

Standard form

(3 1)(2 3) 0xx Factor.

3 1 0 or 2 3 0xx

Zero -factor property.

1.4 -61.4 -6

Example 1USING THE ZERO-FACTOR

PROPERTY

Solve

Solution:

2

6 73xx

3 1 0 or 2 3 0xx

Zero -factor property. 31xor
23x
1 3 x 3 2 x or

Solve each

equation.

1.4 -71.4 -7

Square Root Property

If x 2 = k, then x k or x k

1.4 -81.4 -8

Example 2USING THE SQUARE ROOT

PROPERTY

a.

Solution:

2 17x

By the square root property, the solution set

is 17

Solve each quadratic equation.

1.4 -91.4 -9

Example 2USING THE SQUARE ROOT

PROPERTY

b.

Solution:

2 25x
Since 5.i 1,i the solution set of x 2 =25 is Solve each quadratic equation.

1.4 -101.4 -10

Example 2USING THE SQUARE ROOT

PROPERTY

c.

Solution:

2 ( 4) 12x

Use a generalization of the square root

property. 2 ( 4) 12x 4 12x

Generalized square

root property. 124x

Add 4.

234x

12 4 3 2 3

Solve each quadratic equation.

1.4 -111.4 -11

Solving A Quadratic Equation

By Completing The Square

To solve ax

2 + bx+ c= 0, by completing the square:

If aa.

Rewrite the equation so that the constant term is

alone on one side of the equality symbol. Step 3Square half the coefficient of x, and add this square to both sides of the equation. Step 4Factor the resulting trinomial as a perfect square and combine like terms on the other side. Step 5 Use the square root property to complete the solution.

1.4 -121.4 -12

Example 3USING THE METHOD OF

COMPLETING THE SQUARE a= 1

Solve 2 -4-14 = 0 by completing the square.

Solution

Step 1This step is not necessary since = 1.

Step 2

2 144

Add 14 to both

sides.

Step 3

2

44 144

add 4 to both sides.

Step 4

2 ( 2) 18

Factor; combine

terms. 2

1()44;2

1.4 -131.4 -13

Example 3USING THE METHOD OF

COMPLETING THE SQUARE a= 1

Solve 2 -4-14 = 0 by completing the square.

Solution

Step 4

2 ( 2) 18

Factor; combine terms.

Step 52 18

Square root property.

Take both

roots. 2 18

Add 2.

2 32

Simplify the radical.

The solution set is

2 3 2.

1.4 -141.4 -14

Example 4USING THE METHOD OF

COMPLETING THE SQUARE a

Solve 9

2 -12+ 9 = 0 by completing the square.

Solution

2

9 12 9 0

2 4103

Divide by 9. (Step 1)

2 413

Add-1. (Step 2)

2 44
9 1 394
2

1 44;add 4

32 99

1.4 -151.4 -15

Example 4USING THE METHOD OF

COMPLETING THE SQUARE a= 1

Solve 9

2 -12+ 9 = 0 by completing the square.

Solution

2 44
9 1 394
2

1 44;add 4

32 99
2 25
39

Factor, combine

terms. (Step 4) 25
39

Square root property

1.4 -161.4 -16

Example 4USING THE METHOD OF

COMPLETING THE SQUARE a= 1

Solve 9

2 -12x + 9 = 0 by completing the square.

Solution

25
39

Quotient rule for

radicals 25
33

Square root property

25
33

Add Ҁ

1.4 -171.4 -17

Example 4USING THE METHOD OF

COMPLETING THE SQUARE a= 1

The solution set isSolution

25
33

Add Ҁ

25
.33

Solve 9

2 -12x + 9 = 0 by completing thequotesdbs_dbs17.pdfusesText_23