### Instruction: The quadratic $8x^2+12x-14$ has two real roots. What is the sum of the squares of these roots? Express your answer as a common fraction in lowest terms.### Response: I know that the sum of the roots of a quadratic equation is equal to the opposite of the coefficient of $x$ divided by the coefficient of $x^2$, by Vieta's formulas.
So, the sum of the roots of this equation is $-12/8 = -3/2$.
I also know that the product of the roots of a quadratic equation is equal to the constant term divided by the coefficient of $x^2$, also by Vieta's formulas.
So, the product of the roots of this equation is $-14/8 = -7/4$.
Now, I want to find the sum of the squares of the roots, which I can call $r_1^2 + r_2^2$.
I can rewrite this as $(r_1 + r_2)^2 - 2r_1r_2$, using the formula for expanding a square.
But I already know the values of $r_1 + r_2$ and $r_1r_2$, so I can substitute them in.
I get $(-3/2)^2 - 2(-7/4) = 9/4 + 14/4 = 23/4$.
So, the sum of the squares of the roots is $23/4$.