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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