Answer :
Let’s evaluate the expression [tex]\( c = \sqrt{(x-5)^2-(x+3)^2+16x} \)[/tex] step by step.
1. Expand the squared terms:
- [tex]\((x-5)^2 = x^2 - 10x + 25\)[/tex]
- [tex]\((x+3)^2 = x^2 + 6x + 9\)[/tex]
2. Substitute these expansions into the given expression:
[tex]\[ c = \sqrt{(x^2 - 10x + 25) - (x^2 + 6x + 9) + 16x} \][/tex]
3. Combine like terms inside the square root:
- Distribute the subtraction over the terms:
[tex]\[ (x^2 - 10x + 25) - (x^2 + 6x + 9) = x^2 - 10x + 25 - x^2 - 6x - 9 \][/tex]
- Simplify the expression:
[tex]\[ = x^2 - x^2 - 10x - 6x + 25 - 9 \][/tex]
[tex]\[ = -16x + 16 \][/tex]
- Don’t forget to add the [tex]\(16x\)[/tex] from the original problem:
[tex]\[ -16x + 16 + 16x = 16 \][/tex]
4. Simplify inside the square root:
[tex]\[ c = \sqrt{16} \][/tex]
5. Compute the square root:
[tex]\[ c = 4 \][/tex]
So, the simplified result is [tex]\( c = 4 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
1. Expand the squared terms:
- [tex]\((x-5)^2 = x^2 - 10x + 25\)[/tex]
- [tex]\((x+3)^2 = x^2 + 6x + 9\)[/tex]
2. Substitute these expansions into the given expression:
[tex]\[ c = \sqrt{(x^2 - 10x + 25) - (x^2 + 6x + 9) + 16x} \][/tex]
3. Combine like terms inside the square root:
- Distribute the subtraction over the terms:
[tex]\[ (x^2 - 10x + 25) - (x^2 + 6x + 9) = x^2 - 10x + 25 - x^2 - 6x - 9 \][/tex]
- Simplify the expression:
[tex]\[ = x^2 - x^2 - 10x - 6x + 25 - 9 \][/tex]
[tex]\[ = -16x + 16 \][/tex]
- Don’t forget to add the [tex]\(16x\)[/tex] from the original problem:
[tex]\[ -16x + 16 + 16x = 16 \][/tex]
4. Simplify inside the square root:
[tex]\[ c = \sqrt{16} \][/tex]
5. Compute the square root:
[tex]\[ c = 4 \][/tex]
So, the simplified result is [tex]\( c = 4 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
