Question

solve.
\\(\sqrt{x + 51} + \sqrt{x - 12} = 9\\)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. \\(x = \square\\)
(type an integer or a simplified fraction. use a comma to separate answers as needed.)

b. there is no real solution.

Explanation:

Step1: Isolate one square root

$\sqrt{x+51} = 9 - \sqrt{x-12}$

Step2: Square both sides

$(\sqrt{x+51})^2 = (9 - \sqrt{x-12})^2$
$x+51 = 81 - 18\sqrt{x-12} + x - 12$

Step3: Simplify the equation

$x+51 = x + 69 - 18\sqrt{x-12}$
$51 - 69 = -18\sqrt{x-12}$
$-18 = -18\sqrt{x-12}$

Step4: Solve for remaining square root

$\sqrt{x-12} = 1$

Step5: Square both sides again

$(\sqrt{x-12})^2 = 1^2$
$x - 12 = 1$

Step6: Solve for x

$x = 13$

Step7: Verify the solution

Substitute $x=13$ into original equation:
$\sqrt{13+51} + \sqrt{13-12} = \sqrt{64} + \sqrt{1} = 8 + 1 = 9$, which matches the right-hand side.

Answer:

A. $x=13$