Question

Find the value of the expression $2x^3+2x^2-7x+72$ when $x=\frac{-1+\sqrt{15}}{2}$

Answer 1

It is given that $x=\frac{-1+\sqrt{15}}{2}$ which can be solved as shown below:

$x=\frac{-1+\sqrt{15}}{2}\Rightarrow 2x=-1+\sqrt{15}\Rightarrow 2x+1=\sqrt{15}\Rightarrow (2x+1{)}^2=(\sqrt{15}{)}^2\Rightarrow (2x{)}^2+1^2+(2\times 2x\times 1)=15(\because (a+b{)}^2=a^2+b^2+2ab)\Rightarrow 4x^2+1+4x=15\Rightarrow 4x^2+4x-14=0\Rightarrow 2(2x^2+2x-7)=0\Rightarrow 2x^2+2x-7=0....\left(1\right)$

Now, consider the given polynomial $2x^3+2x^2-7x+72$ as follows:

$2x^3+2x^2-7x+72=x(2x^2+2x-7)+72=x\left(0\right)+72\left(Fromeqn1\right)=(x\times 0)+72=0+72=72$

Hence, $2x^3+2x^2-7x+72=72$ when $x=\frac{-1+\sqrt{15}}{2}$.