1. 5.5 Solving Quadratic Equations by Factoring

2. Solving Quadratic Equations by Factoring.
A quadratic equation is an equation that can be written in the form
ax2 + bx + c = 0,
where a, b, and c are real numbers, with a ≠ 0.
The form ax2 + bx + c = 0 is the standard form of a quadratic equation. For example,
and
are all quadratic equations, but only x2 + 5x +6 = 0 is in standard form.
Until now, we have factored expressions, including many quadratic expressions. In this section we see how we can use factored quadratic expressions to solve quadratic equations.
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3. Solve quadratic equations by factoring.
Objective 1
Solve quadratic equations by factoring.
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4. Solve quadratic equations by factoring.
We use the zero-factor property to solve a quadratic equation by factoring.
Zero-Factor Property
If a and b are real numbers and if ab = 0, then a = 0 or b = 0.
That is, if the product of two numbers is 0, then at least one of the numbers must be 0. One number must, but both may be 0.
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5. Using the Zero-Factor Property
CLASSROOM EXAMPLE 1
Using the Zero-Factor Property
Solve.
Solution: or or
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6. Solving Quadratic Equations
CLASSROOM EXAMPLE 2
Solving Quadratic Equations
Solve each equation.
Solution: or or
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7. Solving a Quadratic Equation by Factoring
Solve quadratic equations by factoring. (cont’d)
Solving a Quadratic Equation by Factoring
Step 1: Write the equation in standard form — that is, with all terms on one side of the equals sign in descending power of the variable and 0 on the other side.
Step 2: Factor completely.
Step 3: Use the zero-factor property to set each factor with variable equal to 0, and solve the resulting equations.
Step 4: Check each solution in the original equation.
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8. CLASSROOM EXAMPLE 3 Solve 3m2 − 9m = 30. Solution:
Solving a Quadratic Equation with a Common Factor
Solve 3m2 − 9m = 30.
Solution:
A common error is to include the common factor 3 as a solution. Only factors containing variables lead to solutions.
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9. Solving Quadratic Equations
CLASSROOM EXAMPLE 4
Solving Quadratic Equations
Solve each equation.
Solution:
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10. Solving Quadratic Equations (cont’d)
CLASSROOM EXAMPLE 4
Solving Quadratic Equations (cont’d)
Solve the equation.
Solution:
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11. Solving Quadratic Equations with Double Solutions
CLASSROOM EXAMPLE 5
Solving Quadratic Equations with Double Solutions
Solve.
Solution:
There is no need to write the same number more than once in a solution set when a double solution occurs.
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12. Solve other equations by factoring.
Objective 2
Solve other equations by factoring.
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13. CLASSROOM EXAMPLE 6 Solve the equation. Solution:
Solving Equations with More than Two Variable Factors
Solve the equation.
Solution:
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14. CLASSROOM EXAMPLE 6 Solve the equation. Solution:
Solving Equations with More Than Two Variable Factors (cont’d)
Solve the equation.
Solution:
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15. CLASSROOM EXAMPLE 7 Solve. Solution:
Solving an Equation Requiring Multiplication before Factoring
Solve.
Solution:
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