# How to find the equation given the roots

Before I answer the question in the title, we need to do some revisions:

This is what you do when you solve a quadratic equation like $x^2-3x-10=0$:

Solution:

=> (x-5)(x+2)=0

=> If x-5 = 0, then x = 5; if x+2 = 0, then x =-2.

The solution is x=5 or x=-2. If you graph the equation, these roots are of course the x-intercepts. If you forgot how to do it, click how to solve quadratic equation by graphing.

If the question is “Find an equation given the roots 5 and -2.”, then all you need to do is to get the product of (x-5)(x+2) and equate to 0. Is this solution unique? I’ll leave it to you find out.

Someone in the Facebook page of this blog, K-12 Math Problems, left me a message:

How do you find a quartic equation given the roots 1, 2, 3, and 5?

Here’s how you do it: (x-1)(x-2)(x-3)(x-5)=0. Just multiply the four binomials and you will get an equation. Is this solution unique? Are there other quartic equations with the same solution?

Here’s the graphical solution. The roots are the intersection of the graph and the x-axis.