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?
Neil's quadratic is x^2 - 7x - 10, and Chris's quadratic is x^2 + 3x - 4. Therefore, the actual quadratic is x^2 + 3x - 10. The roots of x^2 + 3x - 10 = (x - 2)(x + 5) are 2 and -5.
We can use vietas formulas for this problem. First, realize that we can look at the values that each respective person(Neil and Chris) got correct. First since Neil got his value of C correct, let's look at C only. If the roots of x^2 + bx + c are 1 and 6, then by vietas formulas, c = 1 * 6, which means c = 6. Next, let's look at Chris' equation; specifically, his value of b, which was correct. By vietas formulas, -b = -1 + (-4), which gives us -b = -5, and multiplying by -1 on both sides, gives us b = 5. We then get the correct b and c values for the equation, giving us the quadratic:
x^2 -5x + 6. We can immediately factor this into (x -3) (x-2), giving us the roots of 3 and 2