Asked by Bob
Wilma and Greg were trying to solve the quadratic equation
\[x^2 + bx + c = 0.\]Wilma wrote down the wrong value of $b$ (but her value of $c$ was correct), and found the roots to be $-5$ and $12.$ Greg wrote down the wrong value of $c$ (but his value of $b$ was correct), and found the roots to be $8$ and $-12.$ What are the actual roots of $x^2 + bx + c = 0$?
\[x^2 + bx + c = 0.\]Wilma wrote down the wrong value of $b$ (but her value of $c$ was correct), and found the roots to be $-5$ and $12.$ Greg wrote down the wrong value of $c$ (but his value of $b$ was correct), and found the roots to be $8$ and $-12.$ What are the actual roots of $x^2 + bx + c = 0$?
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Answered by
GPT-5 mini
Wilma's polynomial (with incorrect b) has roots -5 and 12, so c = (-5)(12) = -60.
Greg's polynomial (with incorrect c) has roots 8 and -12, so b = -(8 + (-12)) = 4.
Thus the actual equation is x^2 + 4x - 60 = 0, which factors to (x - 6)(x + 10) = 0.
The roots are 6 and -10.
Greg's polynomial (with incorrect c) has roots 8 and -12, so b = -(8 + (-12)) = 4.
Thus the actual equation is x^2 + 4x - 60 = 0, which factors to (x - 6)(x + 10) = 0.
The roots are 6 and -10.
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