To solve the equation [tex]\((x-4)(x+4) = (x+4)(x-7) + 33\)[/tex], let's carefully work through the problem step by step:
1. Expand the expressions:
First, expand both sides of the equation.
- Left-hand side:
[tex]\[
(x-4)(x+4) = x^2 - 16
\][/tex]
- Right-hand side:
[tex]\[
(x+4)(x-7) + 33 = x^2 - 3x - 28 + 33 = x^2 - 3x + 5
\][/tex]
2. Set up the equation:
Now set the expanded forms equal to each other:
[tex]\[
x^2 - 16 = x^2 - 3x + 5
\][/tex]
3. Simplify the equation:
Subtract [tex]\(x^2\)[/tex] from both sides to simplify:
[tex]\[
-16 = -3x + 5
\][/tex]
4. Isolate the variable:
Combine like terms by first moving the constant term from the right-hand side to the left-hand side.
[tex]\[
-16 - 5 = -3x
\][/tex]
Simplify:
[tex]\[
-21 = -3x
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Divide both sides by -3 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-21}{-3} = 7
\][/tex]
6. Verify the solution:
Plug [tex]\(x = 7\)[/tex] back into the original equation to verify:
- Left-hand side:
[tex]\[
(7-4)(7+4) = 3 \times 11 = 33
\][/tex]
- Right-hand side:
[tex]\[
(7+4)(7-7) + 33 = 11 \times 0 + 33 = 33
\][/tex]
Both sides are equal, confirming that [tex]\(x = 7\)[/tex] is a correct solution.
Hence, the correct answer is [tex]\( \boxed{7} \)[/tex].
