Solving Word Problems by Factoring Polynomials - area of a rectangle - Free Printable

Problem Statement:

The length of a rectangle is 8 more than its width. Its area is 240 square feet. Write an equation for its area letting \( w \) be the width, then solve the equation for \( w \) and also find the length.

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Step-by-Step Solution:

#### 1. Define Variables:
Let:
- \( w \): the width of the rectangle (in feet).
- \( w + 8 \): the length of the rectangle (since the length is 8 more than the width).

#### 2. Write the Equation for the Area:
The area of a rectangle is given by:
\[
\text{Area} = \text{length} \times \text{width}
\]
Substitute the given values:
\[
\text{Area} = (w + 8) \cdot w
\]
We are told that the area is 240 square feet, so:
\[
(w + 8) \cdot w = 240
\]

#### 3. Expand and Rearrange the Equation:
Expand the left-hand side:
\[
w(w + 8) = w^2 + 8w
\]
So the equation becomes:
\[
w^2 + 8w = 240
\]
Rearrange this into standard quadratic form (\( ax^2 + bx + c = 0 \)):
\[
w^2 + 8w - 240 = 0
\]

#### 4. Solve the Quadratic Equation by Factoring:
We need to factor the quadratic equation:
\[
w^2 + 8w - 240 = 0
\]
To factor this, we look for two numbers that:
- Multiply to \( -240 \) (the constant term),
- Add up to \( 8 \) (the coefficient of the linear term).

The two numbers that satisfy these conditions are \( 20 \) and \( -12 \), because:
\[
20 \cdot (-12) = -240 \quad \text{and} \quad 20 + (-12) = 8
\]

Using these numbers, we can rewrite the middle term \( 8w \) as \( 20w - 12w \):
\[
w^2 + 20w - 12w - 240 = 0
\]
Now, factor by grouping:
\[
(w^2 + 20w) + (-12w - 240) = 0
\]
Factor out the greatest common factor from each group:
\[
w(w + 20) - 12(w + 20) = 0
\]
Notice that \( (w + 20) \) is a common factor:
\[
(w + 20)(w - 12) = 0
\]

#### 5. Solve for \( w \):
Set each factor equal to zero:
\[
w + 20 = 0 \quad \text{or} \quad w - 12 = 0
\]
Solve each equation:
\[
w + 20 = 0 \implies w = -20
\]
\[
w - 12 = 0 \implies w = 12
\]

Since the width \( w \) must be a positive value, we discard \( w = -20 \). Thus:
\[
w = 12
\]

#### 6. Find the Length:
The length is given by \( w + 8 \):
\[
\text{Length} = w + 8 = 12 + 8 = 20
\]

#### 7. Verify the Solution:
Check the area:
\[
\text{Area} = \text{length} \times \text{width} = 20 \times 12 = 240 \quad \text{(correct)}
\]

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Final Answer:

\[
\boxed{w = 12, \text{ length} = 20}
\]