To solve this problem, let's break it down step by step.
1. Define the variables:
Let [tex]\( s \)[/tex] be the side length of the smaller window in inches.
2. Express the areas:
- The area of the smaller window is [tex]\( s^2 \)[/tex] square inches.
- The side length of the larger window is [tex]\( s + 5 \)[/tex] inches (since the larger window is 5 inches longer on each side).
- The area of the larger window is [tex]\((s + 5)^2\)[/tex] square inches.
3. Total area:
The total area of both windows combined is given as 1,025 square inches.
4. Set up the equation:
[tex]\[
s^2 + (s + 5)^2 = 1025
\][/tex]
5. Expand and simplify:
[tex]\[
s^2 + (s^2 + 10s + 25) = 1025
\][/tex]
Combine like terms:
[tex]\[
2s^2 + 10s + 25 = 1025
\][/tex]
6. Form a quadratic equation:
[tex]\[
2s^2 + 10s + 25 - 1025 = 0
\][/tex]
[tex]\[
2s^2 + 10s - 1000 = 0
\][/tex]
7. Simplify the quadratic equation:
Divide every term by 2:
[tex]\[
s^2 + 5s - 500 = 0
\][/tex]
8. Solve the quadratic equation:
We can solve the quadratic equation [tex]\( s^2 + 5s - 500 = 0 \)[/tex] using the quadratic formula:
[tex]\[
s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -500 \)[/tex].
Plug in these values:
[tex]\[
s = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-500)}}{2 \cdot 1}
\][/tex]
[tex]\[
s = \frac{-5 \pm \sqrt{25 + 2000}}{2}
\][/tex]
[tex]\[
s = \frac{-5 \pm \sqrt{2025}}{2}
\][/tex]
[tex]\[
s = \frac{-5 \pm 45}{2}
\][/tex]
This gives us two potential solutions:
[tex]\[
s = \frac{40}{2} = 20
\][/tex]
[tex]\[
s = \frac{-50}{2} = -25
\][/tex]
9. Interpret the solutions:
Since a side length cannot be negative, we discard [tex]\( s = -25 \)[/tex].
Therefore, the side length of the smaller window is [tex]\( 20 \)[/tex] inches.
