Solving Equations by Factoring

Show your work for each problem.

1.
What is the zero product property?
1. a)If , then or
2. b)If , then or
3. c)If , then
4. d)If , then and
Answer: If , then or
The zero product property states that if the product of two factors equals zero, at least one of the factors must be zero. This is the foundation of solving equations by factoring.
2.
Solve:
1. a) or
2. b) or
3. c) or
4. d) or
Answer: or
From , we get . From , we get . The solution has the opposite sign of what appears in the factor.
3.
Solve . Enter the positive solution only.
Answer: 4
. So or . The positive solution is .
4.
Solve . Enter the non-zero solution.
Answer: -5
. So or , giving . The non-zero solution is .
5.
Solve by factoring.
Answer: x = 2 or x = 3
1. Find two numbers that multiply to 6 and add to -5. What are they? -2 and -3
2. Factor the trinomial using these numbers. (x - 2)(x - 3)
3. Set the first factor equal to zero: . Solve for . 2
4. Set the second factor equal to zero: . Solve for . 3
6.
Solve:
1. a) or
2. b) or
3. c) or
4. d) or
Answer: or
Factor: . The numbers 5 and -3 multiply to -15 and add to 2. So or .
7.
Solve . Enter both solutions separated by a comma (smaller first).
Answer: 2, 5
Rewrite as . Factor: . Solutions: or .
8.
Solve . Enter both solutions separated by a comma (smaller first).
Answer: 0, 4
. So giving , or giving .
9.
Solve by factoring.
Answer: x = 7 or x = -7
1. What pattern does match? difference of squares
2. Factor using (x + 7)(x - 7)
3. Solve -7
4. Solve 7
10.
Solve by factoring.
Answer: x = -4 or x = 6
1. Find two numbers that multiply to -24 and add to -2 4 and -6
2. Factor the trinomial (x + 4)(x - 6)
3. Solve -4
4. Solve 6
11.
Solve:
1. a) or
2. b) or
3. c) or
4. d) or
Answer: or
. Product . Numbers: and (, ). Rewrite: . So or .
12.
Solve . How many distinct solutions are there?
Answer: 1
. So , giving . There is only 1 distinct solution (a repeated root).
13.
Solve by factoring.
Answer: x = -3 or x = -1/2
1. Calculate where and 6
2. Find two numbers that multiply to 6 and add to 7 1 and 6
3. Factor by grouping: . Write the factored form. (x + 3)(2x + 1)
4. Solve -3
5. Solve -1/2
14.
A rectangle has length cm and width cm. If the area is 36 square cm, find the width.
Answer: 4
1. Write the area equation: length times width equals area x(x + 5) = 36
2. Expand and rewrite in standard form x^2 + 5x - 36 = 0
3. Factor the trinomial (x + 9)(x - 4)
4. Since width must be positive, what is ? 4
15.
Why should you never divide both sides of by ?
1. a)You get a wrong answer
2. b)The equation becomes too complicated
3. c)You lose the solution
4. d)Division by is undefined
Answer: You lose the solution
If you divide by , you assume . But IS a valid solution! The correct method: factor as , giving or .