To solve the problem of finding the dimensions of the original square photo, we start with the equation given:
[tex]$(x + 10)^2 = 256$[/tex]
Here, [tex]\( x \)[/tex] represents the side measure of the original square photo, and the equation describes the relationship between the side length of the original photo and the enlarged photo.
Step-by-Step Breakdown:
1. Understand the equation:
The equation [tex]\((x + 10)^2 = 256\)[/tex] suggests that if you add 10 inches to each side of the original square photo, the area of the enlarged photo becomes 256 square inches.
2. Solve for [tex]\( x + 10 \)[/tex]:
To isolate [tex]\( x \)[/tex], we first take the square root of both sides of the equation:
[tex]\[
\sqrt{(x + 10)^2} = \sqrt{256}
\][/tex]
This simplifies to:
[tex]\[
x + 10 = 16 \quad \text{or} \quad x + 10 = -16
\][/tex]
3. Determine valid solution:
Since [tex]\( x \)[/tex] represents a physical length (the side of a square), we discard the negative solution as length cannot be negative. Therefore:
[tex]\[
x + 10 = 16
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Subtract 10 from both sides to find [tex]\( x \)[/tex]:
[tex]\[
x = 16 - 10
\][/tex]
Which gives us:
[tex]\[
x = 6
\][/tex]
5. Verify the solution:
The original side length [tex]\( x \)[/tex] is therefore 6 inches. To confirm this, we can check our work:
[tex]\[
(6 + 10)^2 = 16^2 = 256
\][/tex]
Which is correct, since the area of the enlarged photo indeed turns out to be 256 square inches.
Conclusion:
The dimensions of the original square photo were [tex]\(6\)[/tex] inches by [tex]\(6\)[/tex] inches. Therefore, the correct answer is:
6 inches by 6 inches.
