# What is the sum of the square of the roots of x^2 - 6x + 8 = 0 There's a method that doesn't require you to find the roots themselves. Call the unknown roots `r_1,r_2.` We know that the quadratic factors as

`x^2-6x+8=(x-r_1)(x-r_2),` and expanding the right side gives

`x^2-6x+8=x^2-(r_1+r_2)x+r_1r_2.` Equating coefficients gives

`r_1+r_2=6`                   (1)

`r_1r_2=8`                      (2)

Multiply equation (1) by `r_1` and then `r_2` to get the new equations

`r_1^2+r_1r_2=6r_1`           (3)

`r_1r_2+r_2^2=6r_2`           (4)

Add (3) and (4) to get `r_1^2+r_2^2+2r_1r_2=6(r_1+r_2).`

Now use (1) and (2) to get `r_1^2+r_2^2+16=36,` so we get the answer `r_1^2+r_2^2=20.`

For this particular problem, this method takes longer. However, it can be used to solve similar problems where solving for the roots is tedious.

Approved by eNotes Editorial Team The quadratic equation given is x^2 - 6x + 8 = 0.

x^2 - 6x + 8 = 0

=> x^2 - 4x - 2x + 8 = 0

=> x(x - 4) - 2(x - 4) = 0

=> (x - 2)(x - 4) = 0

=> x = 2 and x = 4

The sum of the squares of the roots is 2^2 + 4^2 = 4 + 16 = 20

The required sum is 20.