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 1 and 6. Greg 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?

The roots are - 2 and - 3

Step-by-step explanation:

For a quadratic equation given as x² + bx + c, for roots x and y, the sum of roots is equal to the negation of the coefficient of the second term (i. e x + y = - b) while the products of the roots is equal to the coefficient of the second term (i. e x * y = c).

Since for the roots 1 and 6, only the value of c was correct, to get c we use the product of roots. Therefore, c = 1 * 6 = 6

Since for the roots - 1 and - 6, only the value of b was correct, to get b we use the sum of roots. Therefore:

-b = - 1 + - 4 = - 5

b = 5

Since the quadratic equation is x² + bx + c, substituting value of b and c and solving:

x² + 5x + 6 = 0

x² + 2x + 3x + 6 = 0

x (x + 2) + 3 (x + 2) = 0

(x + 2) (x + 3) = 0

x = - 2 or x = - 3

The roots are - 2 and - 3