Solving Equations by Factoring

Definition of Quadratic EquationsZero-Factor PropertyStrategy for Solving Quadratics

Standard Form Quadratic Equation

Quadratic equations can be written in the form

ax2 + bx + c = 0where a, b, and c are real numbers with a 0.

Standard form for a quadratic equation is in descending order equal to zero.

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Examples of Quadratic Equations

pp 18812

1892 xx

252 y

081182 pp

Standard Form

01892 xx

0252 y

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Zero-Factor Property

If a and b are real numbers

and if ab =0, then

a = 0 or

b = 0.

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Solve the equation (x + 2)(2x - 1)=0

• By the zero factor property we know...

• Since the product is equal to zero then one of the factors must be zero.

0)2( x2x

OR (2 1) 0x 12 x

2

1

2

2x 2

1x

}2

1,2{x

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Solve the equation. Check your answers.

0)2)(5( xx

5xOR 02 x

2x { 2, 5}x 05 x Solution Set

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Solve each equation. Check your answers.

0)35( xx

0x OR

035 x

5

3x

}0,5

3{

x

0x

Solution Set

35 x

5

3

5

5 x

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Solving a Quadratic Equation by Factoring

Step 1 Write the equation in standard form.

Step 2 Factor completely.Step 3 Use the zero-factor property.

Set each factor with a variable equal to zero.

Step 4 Solve each equation produced in step 3.

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Solve.

1892 xx01892 xx

0)3)(6( xx

}3,6{xBACK

Solve.

072 xx0)7( xx

}7,0{xBACK

Number Of Solutions

• The degree of a polynomial is equal to the number of solutions.

xxx 32 23 Three solutions!!!

Example

x (x + 1)(x – 3) = 0Set each of the three factors equal to 0.

x = 0 x + 1 = 0 x = -1

x – 3 = 0

x = 3

Solve the resulting equations.

Write the solution set. x = {0, -1, 3}

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Solve the following equations.

1. x2 – 25 = 0

2. x2 + 7x – 8 = 0

3. x2 – 12x + 36 = 0

4. c2 – 8c = 0

x x 5 5 0

x x 8 1 0

x x 6 6 0

c c( ) 8 0

5,5

8,1

6

8,0

1. Get a value of zero on one side of the equation.

2. Factor the polynomial if possible.

3. Apply the zero product property by setting each factor equal to zero.

4. Solve for the variable.

Solving Equations by Factoring

Definition of Quadratic EquationsZero-Factor PropertyStrategy for Solving Quadratics