+1932 Let $a$ and $b$ be the roots of the quadratic equation $2x^2+6x-14=x^2-8x+2$. What is the value of $(2a-3)(4b-6)$?
+189 First, simplify the quadratic into its standard form. This will allow you to identify information about the roots of the equations.
2x2+6x−14=x2−8x+2x2+14x−16=0
We can also expand the product of the two binomials. This will be important shortly, as you will see.
(2a−3)(4b−6)=8ab−12a−12b+18=8ab−12(a+b)+18
We can use Vieta's formula to find the product of the roots and the rum of the roots. For a quadratic, the product of the roots is the constant term, and the sum of the roots is the opposite of the coefficient of the x-term.
ab=−16,a+b=−148ab−12(a+b)+18=8∗−16−12∗−14+18=−128+168+18=58
