For the function $f(x)=6 \sqrt{2 x-8}-6 \sqrt{x+5}$, find all values $x$ for which $f(x)=-18$.
A) 4
B) -11
C) 18
D) 11

Question

Quizplus · Accepted Answer

The given function is $f(x)=6 \sqrt{2 x-8}-6 \sqrt{x+5}$. To find the values of $x$ for which $f(x)=-18$, we set the function equal to -18 and solve for $x$:$$6 \sqrt{2 x-8}-6 \sqrt{x+5} = -18$$Divide both sides by 6:$$\sqrt{2 x-8}-\sqrt{x+5} = -3$$Move the square root term to the other side:$$\sqrt{2 x-8} = \sqrt{x+5} - 3$$Square both sides to eliminate the square roots:$$(\sqrt{2 x-8})^2 = (\sqrt{x+5} - 3)^2$$$$2x - 8 = (x + 5) - 6\sqrt{x+5} + 9$$Simplify:$$2x - 8 = x + 14 - 6\sqrt{x+5}$$Move terms involving $x$ to one side:$$x = 22 + 6\sqrt{x+5}$$Since the original equation involves square roots, and we squared both sides, we need to check any solution in the original equation to ensure it doesn't come from squaring a negative number (which could introduce extraneous solutions). Plugging $x=4$ back into the original equation:$$f(4) = 6 \sqrt{2(4)-8}-6 \sqrt{4+5} = 6(0) - 6(3) = -18$$Thus, $x=4$ is the correct solution, and it satisfies the original equation.

For the Function f(x)=62x−8−6x+5f(x)=6 \sqrt{2 x-8}-6 \sqrt{x+5}f(x)=62x−8​−6x+5​ , Find All Values xxx

For the Function $f(x)=6 \sqrt{2 x-8}-6 \sqrt{x+5}$ , Find All Values $x$