1. Solving Quadratic Equations
…by Factoring

2. Solving Quadratic Equations by Factoring
Get ZERO on one side by itself.
Factor. Consider Common Factors FIRST!
Set each factor = 0.
Solve each part.

3. Solve the second equation.
Solve by factoring.
Original equation
Add 4x to each side.
Factor the binomial.
Zero Product Property or
Solve the second equation.
Answer: The solution set is {0, –4}.
Example 3-1a

4. Check Substitute 0 and –4 in for x in the original equation.
Example 3-1a

5. Subtract 5x and 2 from each side.
Solve by factoring.
Original equation
Subtract 5x and 2 from each side.
Factor the trinomial.
Zero Product Property or
Solve each equation.
Answer: The solution set is Check each solution.
Example 3-1a

6. Solve each equation by factoring. a.
Answer: {0, 3}
Answer:
Example 3-1b

7. Answer: The solution set is {3}.
Solve by factoring.
Original equation
Add 9 to each side.
Factor.
Zero Product Property or
Solve each equation.
Answer: The solution set is {3}.
Example 3-2a

8. Check The graph of the related function, intersects the x-axis only once. Since the zero of the function is 3, the solution of the related equation is 3.
Example 3-2a

9. Solve by factoring.
Answer: {–5}
Example 3-2b

10. Multiple-Choice Test Item What is the positive solution of the equation ?
B 5
C 6
D 7
Read the Test Item
You are asked to find the positive solution of the given quadratic equation. This implies that the equation also has a solution that is not positive. Since a quadratic equation can either have one, two, or no solutions, we should expect this equation to have two solutions.
Example 3-3a

11. Solve the Test Item Original equation Factor. Divide each side by 2.
Zero Product Property
Solve each equation.
Both solutions, –3 and 7, are listed among the answer choices. However, the question asks for the positive solution, 7.
Answer: D
Example 3-3a

12. Multiple-Choice Test Item What is the positive solution of the equation ?
B –5
C 2
D 6
Answer: C
Example 3-3b

13. Write a quadratic equation with and 6 as its
roots. Write the equation in the form
where a, b, and c are integers.
Write the pattern.
Replace p with and q with 6.
Simplify.
Example 3-4a

14. Multiply each side by 3 so that b is an integer.
Use FOIL.
Multiply each side by 3 so that b is an integer.
Answer: A quadratic equation with roots and 6
and integral coefficients is
You can check this result by graphing the
related function.
Example 3-4a

15. Write a quadratic equation with and 5 as its
roots. Write the equation in the form
where a, b, and c are integers.
Answer:
Example 3-4b