Question

Let r1 and r2 be the roots of the quadratic equation x2 - 12x + 9 = 0. Determine the value of ????1 2 + ????2 2 . (Hint: In the first recitation the relationship between the coefficients in a quadratic and the sum and product of the roots of the quadratic was proven. Use this relationship to solve the problem without finding either root.)

Answer 1

The value of
`r_(1)^2+r_(2)^2` is 126

Explanation:

We can use the definition of Vieta's formula for quadratics:

Given
`f(x) = ax^(2) +bx+c`, if the equation f(x) = 0 has roots
`r_(1)` and
`r_(2)` then

`r_(1)+r_(2)=-(b)/(a) , r_(1)\cdot r_(2)=(c)/(a)`

So suppose
`r_(1)` and
`r_(2)` are the roots of the equation
`x^(2) - 12x + 9` to find
`r_(1)^2+r_(2)^2`, note that from our Vieta's formula for quadratics we have

`r_(1)+r_(2)=-(-12)/(1)\\r_(1)+r_(2)=12` and
`r_(1)\cdot r_(2)=(9)/(1)\\r_(1)\cdot r_(2)=9`

Therefore

`r_(1)^2+r_(2)^2=(r_(1)+r_(2))^2-2\cdot r_(1)r_(2)\\r_(1)^2+r_(2)^2=(12)^2-2\cdot 9\\r_(1)^2+r_(2)^2=126`