What two numbers are the product of 9 and the sum of -7

Answer 1

I take it that you're asking which two numbers can be multiplied together to give nine and summed together to give -7. Let's call these numbers a and b. Consider the quadratic polynomial

f(x) = (x+a)(x+b) = x^2 +(a+b)x + ab

The roots of this equation (values of x for which f(x) = 0) are x = -a and x = -b. It follows that if we set a+b = -7 and ab = 9 and solve the equation f(x) = 0, then the opposites of these solutions will give us the numbers we seek. We thus solve

x^2 - 7x + 9 = 0

for x using the quadratic formula. This gives

x* = [7 +/- sqrt(49-36)]/2 = [7 +/- sqrt(13)]/2

By the logic we outlined above, the opposites of these numbers, namely -x* = (-7 +/- sqrt(13))/2, should sum to -7 and have a product of 9. We can check this:

(-7 + sqrt(13))/2 + (-7 - sqrt(13))/2 = -14/2 = -7

(-7 + sqrt(13))/2 * (-7 - sqrt(13))/2 = (49 - 13)/4 = 9