Let's solve the problem step by step.
---
Problem Statement:
The length of a rectangle is two feet less than 3 times the width. If the area is 65 ft², find the dimensions.
We are given:
- Let $ x $ = width
- Then length = $ 3x - 2 $
- Area = length × width = $ x(3x - 2) = 65 $
---
Step 1: Set up the equation
$$
x(3x - 2) = 65
$$
Distribute:
$$
3x^2 - 2x = 65
$$
Bring all terms to one side:
$$
3x^2 - 2x - 65 = 0
$$
---
Step 2: Solve the quadratic equation
We solve:
$$
3x^2 - 2x - 65 = 0
$$
Use the
quadratic formula:
$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$
Where $ a = 3 $, $ b = -2 $, $ c = -65 $
$$
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(-65)}}{2(3)}
$$
$$
x = \frac{2 \pm \sqrt{4 + 780}}{6}
$$
$$
x = \frac{2 \pm \sqrt{784}}{6}
$$
$$
\sqrt{784} = 28
$$
$$
x = \frac{2 \pm 28}{6}
$$
Now calculate both possibilities:
1. $ x = \frac{2 + 28}{6} = \frac{30}{6} = 5 $
2. $ x = \frac{2 - 28}{6} = \frac{-26}{6} = -\frac{13}{3} $ → Not valid (width can't be negative)
So, $ x = 5 $
---
Step 3: Find the length
Length = $ 3x - 2 = 3(5) - 2 = 15 - 2 = 13 $
---
✔ Final Answer:
-
Width = $ 5 $ ft
-
Length = $ 13 $ ft
Check: Area = $ 5 \times 13 = 65 $ ft² ✔️
---
✔ Answer: The dimensions are
5 feet (width) and
13 feet (length).
Parent Tip: Review the logic above to help your child master the concept of factoring polynomials word problems worksheet.
