Question

Solve for the largest value of $x$ such that $5(9x^2+9x+10) = x(9x-40).$ Express your answer as a simplified common fraction.

Answer 1

Hello, please consider the following.

$\begin{aligned}5(9x^2+9x+10) &= x(9x-40)\\&=9x^2-40x\end{aligned}\\\\ 45x^2+45x+50=9x^2-40x\\\\ (45-9)x^2+(45+40)x+50=0\\\\ 36x^2+85x+50=0$

We can estimate the discriminant, and then, the solutions and we take the largest one.

$\Delta=b^2-4ac=85^2-4*36*50=25=5^2\\\\x_1=\dfrac{-85-5}{2*36}=\dfrac{-18*5}{18*4}=\dfrac{-5}{4}\\\\x_2=\dfrac{-85+5}{2*36}=\dfrac{-80}{72}=\dfrac{-8*10}{8*9}=\boxed{\dfrac{-10}{9}}$

Thank you