# How can I find the equation of a parabola only given it's x-intercepts?

I received a problem in my math class the other day that left me stumped. The problem went something like this.

Mr. Lots-O-Cash would like to order a parabola that passes through the points $(-4, 0)$, and $(2, 0)$. Create an equation for this so you can pass the information onto the manufacturing department.

I'm wondering, how can I find the equation for this parabola. It seems like to little info to find one. Please don't give me an answer, just a way to solve it.

• There won't be a unique solution. Think about what the $x$-intercepts tell you about the linear factors of the quadratic expression involved. – David Mitra Jan 13 '15 at 18:58
• $y=c(x+4)(x-2)$ for some $c\neq 0$. – Bernard Jan 13 '15 at 19:01

The general form of a parabola/quadratic is:

$$y=ax^2+bx+c$$

You should be able to construct a pair of simultaneous equations using the two points and solve for $a,b,c$.

Another method is to consider:

$$y=\lambda(x-\alpha)(x-\beta)$$

Where $\alpha , \beta$ are the roots of the parabola.

• What is the value of $\lambda$ here? – Ethan Bierlein Jan 13 '15 at 19:24
• Does the value of $\lamda$ affect the values of the roots? i.e. If you sub the points into the second equation, does it matter what $\lamda$ is, apart from the fact that it can't be 0? – Gridley Quayle Jan 13 '15 at 19:28