Factoring Word Problems

Show your work for each problem.

1.
A rectangular field has length meters and width meters. Which expression represents its area?
1. a)
2. b)
3. c)
4. d)
Answer:
Area = . This is how area problems create quadratic expressions.
2.
If represents the first of two consecutive integers, what represents their product?
1. a)
2. b)
3. c)
4. d)
Answer:
Consecutive integers are and . Their product is , which is quadratic.
3.
A square has side length cm. If the area is 49 square cm, what is ?
Answer: 7
factors as , giving or . Since length must be positive, cm.
4.
The product of two consecutive positive integers is 30. What is the smaller integer?
Answer: 5
factors as . So or . Since we need positive integers, . Check: ✓
5.
A ball's height is given by . Which factored form helps find when it hits the ground?
1. a)
2. b)
3. c)
4. d)
Answer:
. Setting this equal to 0: gives (launch) or (landing).
6.
A rectangular pool has length 4 meters more than its width. The area is 45 square meters. Find the dimensions.
Answer: 5, 9
1. If width is , what is the length? x + 4
2. Write the area equation: 45
3. Expand to get . Rearrange to standard form: x^2 + 4x - 45 = 0
4. Factor: Find numbers that multiply to -45 and add to 4 9 and -5
5. So . What is the width (positive solution)? 5
7.
One number is 3 less than another. Their product is 54. What is the larger number?
Answer: 9
factors as . So or . The larger positive number is 9, the smaller is 6. Check: ✓
8.
A rocket's height is meters after seconds. When does it return to the ground?
Answer: 8
1. What is when the rocket hits the ground? 0
2. Set up the equation: 0
3. Factor out the GCF from 5t
4. After factoring: . What's inside? 8 - t
5. Using , when does it land (not )? 8
9.
Which equation represents: 'The product of two consecutive even integers is 168'?
1. a)
2. b)
3. c)
4. d)
Answer:
Consecutive even integers: and . Their product: . This gives , which factors to , so (the integers are 12 and 14).
10.
A rectangular garden is twice as long as it is wide. If the area is 72 square meters, what is the width?
Answer: 6
(positive value). Width = 6 m, Length = 12 m. Check: ✓
11.
A rectangular photo is 6 cm by 8 cm. It has a frame with a uniform border. If the total area is 120 square cm, find the border width.
Answer: 2
1. If border width is , total width is 2x
2. Total length is 2x
3. Write the area equation: 120
4. Expand: . Simplify to . Rearrange: 4x^2 + 28x - 72 = 0
5. Divide by 4: . Factor: 2
6. What is the border width (positive solution)? 2
12.
The sum of a number and its square is 72. Find the positive number.
Answer: 8
factors as . So or . The positive number is 8. Check: ✓
13.
A ball is thrown upward with height . At what time is the ball at height 32?
1. a) and seconds
2. b) seconds
3. c) seconds
4. d) seconds
Answer: and seconds
. The ball is at height 32 at both (going up) and (coming down).
14.
The product of two consecutive odd integers is 143. Find the integers.
Answer: 11, 13
1. Consecutive odd integers differ by how much? 2
2. If the first odd integer is , the second is: x + 2
3. Write the product equation: x(x + 2) = 143
4. Expand and rearrange to standard form: x^2 + 2x - 143 = 0
5. Factor: Find numbers that multiply to -143 and add to 2 13 and -11
6. What is the first (smaller positive) odd integer? 11
15.
A rectangular courtyard is 5 meters longer than it is wide. A 2-meter wide walkway surrounds it. If the total area (courtyard + walkway) is 204 square meters, what is the width of the courtyard?
Answer: 8
. Width = 8 m, Length = 13 m. Check: ✓