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What is the sum of the square of the roots of x^2 - 6x + 8 = 0
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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.
Posted by justaguide on July 2, 2013 at 2:05 PM (Answer #1)
High School Teacher
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.
Posted by degeneratecircle on July 2, 2013 at 2:43 PM (Answer #2)
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