# How do I evaluate this function?

I have the following problem

$$f(x)=\frac{(x+4)^{\frac12}(2x-x^2)(-\frac12(x+4)-^\frac{1}{2})}{x+4}, x>-4$$

Here's how far I got. I actually went a little farther, but after a couple extra steps I wasn't going anywhere.

1. Rewording the problem and including a constraint $$\frac{\sqrt{x+4}(2x-x^2)-\frac12(\sqrt{x+4})}{x+4}, x>-4$$
2. Creating an equality, multiplying both sides with the denominator, then subtracting $$(2x-x^2)$$ to the other side to square both sides $$(\sqrt{x+4}-\frac12\sqrt{x+4})^2=(-(2x-x^2))^2$$
3. Ending up with this, unable to continue $$\frac52x+10=(5x^3+4x^2-\frac32x+6)^2$$

I'm guessing I probably made some mistake from the start, and I'm guessing I need to work on simplifying roots and exponents better, like, instead of writing $$x^2$$ I write $$xx$$, I know things like that makes factoring, for example, easier. I'm still working on that.

Anyway, I'd appreciate some help here.

UPDATE: My mistake, I overlooked the directions. Apparently, I am to "write the expression as a single quotient in which only positive exponents and/or radicals appear"

• What is the equation you are trying to solve? – Robert Z Feb 20 '19 at 7:55
• @Robert Z, I included those parentheses because all terms are being multiplied together. I wanted to avoid writing $\times$ to indicate multiplication, so I will be including those parentheses again – Lex_i Feb 20 '19 at 7:56
• @RobertZ I'm trying to solve the equation that is quoted – Lex_i Feb 20 '19 at 7:58
• In the statement you wrote a function not an equation. No equal sign!! – Robert Z Feb 20 '19 at 7:59
• BTW the rewording in 1) is not equivalent to the statement... – Robert Z Feb 20 '19 at 8:00

$$\sqrt{x+4}\cdot\sqrt{x+4}=x+4,$$ which for your previous problem gives $$f(x)=\frac{\sqrt{x+4}(2x-x^2)\left(-\frac{1}{2}\sqrt{x+4}\right)}{x+4}=\frac{x^2-2x}{2}$$ and for your new problem gives:
$$f(x)=\frac{\sqrt{x+4}(2x-x^2)\left(-\frac{1}{2}\cdot\frac{1}{\sqrt{x+4}}\right)}{x+4}=\frac{x^2-2x}{2(x+4)}.$$
• Based on your answer, does this mean $x=2$? Because I went for another attempt and got $-\frac{(x+4)\frac12(2x-x^2)}{x+4}$ which led me to $x(\frac12x-1)=0, x=2$ – Lex_i Feb 20 '19 at 8:29