Neil and Chris were trying to solve the quadratic equation
\[x^2 + bx + c = 0.\]Neil wrote down the wrong value of $b$ (but his value of $c$ was correct), and found the roots to be $1$ and $6.$ Chris wrote down the wrong value of $c$ (but his value of $b$ was correct), and found the roots to be $-1$ and $-4.$ What are the actual roots of $x^2 + bx + c = 0$?

Let be the roots of the equation .

The coefficients and the roots are in the following relation:

So, since Neil got the correct value of , we know that the product of his roots is correct: the product of the roots must be 6.

Similarly, since Chris got the correct value of , we know that the opposite of the sum of his roots is correct: the opposite of the sum of the roots must be 5.

So, we want two numbers that give -5 when summed, and 6 when multiplied. Those numbers are -2 and -3.

b

step-by-step explanation:

b

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