Question

The product of the roots of $2x^2 - 7x + N = 0$ is $-5.$ Find $N.$

Answer 1

By the fundamental theorem of algebra, we can factorize the quadratic as

$2x^2-7x+N=2(x-r_1)(x-r_2)$

so that expanding the right hand side gives

$2x^2-7x+N=2x^2-2(r_1+r_2)x+2r_1r_2$

$\implies\begin{cases}-2(r_1+r_2)=-7\\2r_1r_2=N\end{cases}$

We're told that the product of the roots is -5, or $r_1r_2=-5$, so we get

$2(-5)=N=-10$