Answer :
To solve the problem of finding the dimensions of the bedroom, we can use the information given about the relationship between its length and width, and the area of the room.
1. Define Variables:
- Let's say the width of the bedroom is [tex]\( w \)[/tex] feet.
- According to the problem, the length of the bedroom is 2 feet more than the width. So, the length would be [tex]\( w + 2 \)[/tex] feet.
2. Set Up the Equation:
- The formula for the area of a rectangle is given by:
[tex]\[
\text{Area} = \text{length} \times \text{width}
\][/tex]
- Substitute the expressions for the length and width into the formula:
[tex]\[
120 = (w + 2) \times w
\][/tex]
3. Solve the Equation:
- Expand the expression:
[tex]\[
120 = w^2 + 2w
\][/tex]
- Rearrange it to form a quadratic equation:
[tex]\[
w^2 + 2w - 120 = 0
\][/tex]
4. Factor or Use the Quadratic Formula:
- This quadratic equation can be solved using the quadratic formula:
[tex]\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -120 \)[/tex].
- Calculate the discriminant:
[tex]\[
b^2 - 4ac = 2^2 - 4 \times 1 \times (-120) = 4 + 480 = 484
\][/tex]
- Find the square root of the discriminant, which is [tex]\( \sqrt{484} = 22 \)[/tex].
5. Find the Solutions:
- Substitute into the quadratic formula:
[tex]\[
w = \frac{-2 \pm 22}{2}
\][/tex]
- Calculate the two possible values for [tex]\( w \)[/tex]:
[tex]\[
w_1 = \frac{-2 + 22}{2} = \frac{20}{2} = 10
\][/tex]
[tex]\[
w_2 = \frac{-2 - 22}{2} = \frac{-24}{2} = -12
\][/tex]
- Since a width cannot be negative, use the positive value: [tex]\( w = 10 \)[/tex].
6. Calculate the Length:
- The length is [tex]\( w + 2 = 10 + 2 = 12 \)[/tex].
Therefore, the width of the bedroom is 10 feet, and the length is 12 feet.
1. Define Variables:
- Let's say the width of the bedroom is [tex]\( w \)[/tex] feet.
- According to the problem, the length of the bedroom is 2 feet more than the width. So, the length would be [tex]\( w + 2 \)[/tex] feet.
2. Set Up the Equation:
- The formula for the area of a rectangle is given by:
[tex]\[
\text{Area} = \text{length} \times \text{width}
\][/tex]
- Substitute the expressions for the length and width into the formula:
[tex]\[
120 = (w + 2) \times w
\][/tex]
3. Solve the Equation:
- Expand the expression:
[tex]\[
120 = w^2 + 2w
\][/tex]
- Rearrange it to form a quadratic equation:
[tex]\[
w^2 + 2w - 120 = 0
\][/tex]
4. Factor or Use the Quadratic Formula:
- This quadratic equation can be solved using the quadratic formula:
[tex]\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -120 \)[/tex].
- Calculate the discriminant:
[tex]\[
b^2 - 4ac = 2^2 - 4 \times 1 \times (-120) = 4 + 480 = 484
\][/tex]
- Find the square root of the discriminant, which is [tex]\( \sqrt{484} = 22 \)[/tex].
5. Find the Solutions:
- Substitute into the quadratic formula:
[tex]\[
w = \frac{-2 \pm 22}{2}
\][/tex]
- Calculate the two possible values for [tex]\( w \)[/tex]:
[tex]\[
w_1 = \frac{-2 + 22}{2} = \frac{20}{2} = 10
\][/tex]
[tex]\[
w_2 = \frac{-2 - 22}{2} = \frac{-24}{2} = -12
\][/tex]
- Since a width cannot be negative, use the positive value: [tex]\( w = 10 \)[/tex].
6. Calculate the Length:
- The length is [tex]\( w + 2 = 10 + 2 = 12 \)[/tex].
Therefore, the width of the bedroom is 10 feet, and the length is 12 feet.
