+1791 Let $a$ and $b$ be the roots of the quadratic equation $2x^2+6x-14=x^2-8x+2$. What is the value of $(2a-3)(4b-6)$?
+177 First, simplify the quadratic into its standard form. This will allow you to identify information about the roots of the equations.
\(2x^2 + 6x - 14 = x^2 - 8x + 2 \\ x^2 + 14x - 16 = 0\)
We can also expand the product of the two binomials. This will be important shortly, as you will see.
\(\begin{align*} (2a - 3)(4b - 6) &= 8ab - 12a - 12b + 18 \\ &= 8ab - 12(a + b) + 18 \end{align*}\)
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.
\(ab = -16, a + b = -14 \\ \begin{align*} 8ab - 12(a + b) + 18 &= 8 * -16 - 12 * -14 + 18 \\ &= -128 + 168 + 18 \\ &= 58 \end{align*} \)
