Hi, there--

THE PROBLEM:
Write a quadratic equation having the given numbers as solutions: -9 and -6.

A SOLUTION:
A quadratic equation is an equation with an x-squared term. It has at most two real roots. To find an 
equation that goes with the roots -9 and -6, we will "work backwards."

If I had a quadratic equation and want to find its roots, I would factor the equation and solve each of the 
factors. That's "working frontwards". I can also work backwards from the solutions. 

The factors we find by working backwards always have the form (x - a).

Having a factor of (x - a) means the same thing as having a solution of x = a. In other words, if "x – a" is 
a factor, then "x = a" is a solution, and vice versa. We use this fact to find quadratics from their roots.

We have a quadratic with two solutions: -9 and -6. This implies that x = -9 OR x = -6

If x = -9, then it came from the factor equation x + 9 = 0
If x = -6, then it came from the factor equation x + 6 = 0

This means that (x + 9) and (x + 6) are the factors of the quadratic

Remember, quadratic can have at most two solutions, so these are the only factors.

Therefore, the original quadratic in factored form was something like:
y = (x + 9)(x + 6)

You can leave this in factored form or multiply it out.
y = x^2 + 15x + 54


By the way, there are many, many other quadratic equations that have these two solutions. For
example, suppose you have the equation, y = 3(x + 9)(x + 6) We still have the same two solutions, 
because the factor 3 does not yield a root. 

You can read more about this here: http://www.purplemath.com/modules/fromzero.htm


Hope this helps! Feel free to email if you have any questions about the solution.

Good luck with your math,

Mrs. F
math.in.the.vortex@gmail.com