Question

Solve the equation without the use of a calculator. please Help!!! Thank you!!! 12x^3-30x=5-2x^2

Answer 1

So, since we have a cubic equation with 4 terms, the first thing we should try is factoring by grouping, so:

$12x^3+2x^2-30x+5 =0 \implies \\ 2x^2(6x+1)-5(6x+1)=0 \implies \\ (2x^2-5)(6x+1) = 0$

Now that we've factored our equation, we can use ZPP and break it up:

$2x^2-5 = 0 \implies \\ 2x^2=5 \implies\\ x^2=\frac{5}{2} \implies\\ x=\pm\frac{\sqrt{5}}{\sqrt{2}}=\pm\frac{\sqrt{10}}{2} \\ \\ 6x+1 =0 \implies \\ 6x=-1 \implies\\ x=\frac{-1}{6}$

So, our solutions are:

$x\in \{\frac{\sqrt{10}}{2},-\frac{\sqrt{10}}{2},-\frac{1}{6}\}$