To find the length of the sides of the smaller window given that the total area of two square windows is 1025 square inches, and that each side of the larger window is 5 inches longer than the sides of the smaller window, let’s follow these steps:
1. Define the variables:
- Let [tex]\( x \)[/tex] be the side length of the smaller window.
- Therefore, the side length of the larger window will be [tex]\( x + 5 \)[/tex] inches.
2. Write the equations for the areas:
- The area of the smaller window is [tex]\( x^2 \)[/tex].
- The area of the larger window is [tex]\( (x + 5)^2 \)[/tex].
3. Express the total area:
- The total area of both windows together is given by:
[tex]\[
x^2 + (x + 5)^2 = 1025
\][/tex]
4. Expand and simplify the equation:
- Expand the term [tex]\( (x + 5)^2 \)[/tex]:
[tex]\[
(x + 5)^2 = x^2 + 10x + 25
\][/tex]
- Thus, the equation becomes:
[tex]\[
x^2 + x^2 + 10x + 25 = 1025
\][/tex]
- Combine like terms:
[tex]\[
2x^2 + 10x + 25 = 1025
\][/tex]
- Subtract 1025 from both sides to set the equation to zero:
[tex]\[
2x^2 + 10x - 1000 = 0
\][/tex]
5. Solve the quadratic equation:
- The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex].
- Here, [tex]\( a = 2 \)[/tex], [tex]\( b = 10 \)[/tex], and [tex]\( c = -1000 \)[/tex].
6. Calculate the discriminant:
- The discriminant [tex]\( \Delta \)[/tex] of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
- Substituting the values:
[tex]\[
\Delta = 10^2 - 4 \cdot 2 \cdot (-1000) = 100 + 8000 = 8100
\][/tex]
7. Find the roots using the quadratic formula:
- The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
- Plugging in the values:
[tex]\[
x = \frac{-10 \pm \sqrt{8100}}{4}
\][/tex]
8. Calculate the two possible solutions for [tex]\( x \)[/tex]:
- [tex]\[
x_1 = \frac{-10 + 90}{4} = \frac{80}{4} = 20
\][/tex]
- [tex]\[
x_2 = \frac{-10 - 90}{4} = \frac{-100}{4} = -25
\][/tex]
9. Select the valid solution:
- Since a side length cannot be negative, the valid solution is [tex]\( x = 20 \)[/tex].
Therefore, the side of the smaller window is 20 inches.
The correct answer is: 20 in.
