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