Express in simplest form: 3x+184x+8÷x2−36x+2 .(1 point)

Question

Responses

14(x−2)
Start Fraction 1 over 4 left parenthesis x minus 2 right parenthesis End Fraction

34x−24
Start Fraction 2 over 4 x minus 24 End Fraction

4(x−6)3
Start Fraction 4 left parenthesis x minus 6 right parenthesis over 3 End Fraction

34x−8

Answer 1

To simplify the expression $\frac{3x+184x+8}{x^2-36x+2}$, we first combine like terms in the numerator:

1. Combine the terms in the numerator: $$3x+184x+8=187x+8$$ So, the expression becomes: $$\frac{187x+8}{x^2-36x+2}$$

2. Next, we will factor the denominator $x^2-36x+2$. To factor it, we look for two numbers that multiply to $2$ and add up to $-36$. This doesn't factor nicely with integers, so we will use the quadratic formula to find the roots.

   The roots are given by: $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}=\frac{36\pm \sqrt{(-36{)}^2-4\cdot 1\cdot 2}}{2\cdot 1}$$ $$=\frac{36\pm \sqrt{1296-8}}{2}=\frac{36\pm \sqrt{1288}}{2}=\frac{36\pm 2\sqrt{322}}{2}=18\pm \sqrt{322}$$

   Since we cannot simplify the quadratic further nice integers or simpler rational factors, we will then leave the denominator unsimplified.

3. Therefore, we have $\frac{187x+8}{x^2-36x+2}$, which cannot be simplified further in a simple manner.

None of the given options seem to match this result. Please review the original expression to ensure that there are no errors in the terms provided. If the original expression or values need adjustment to match one of the options given (14(x−2), $\frac{1}{4}(x-2)$, $\frac{34x-24}{4x-24}$, or $\frac{4(x-6)}{3}$), please adjust as necessary for me to give a more accurate simplification.