How do you write the quadratic equation whose roots are n and -3n?

Jan 11, 2017

${x}^{2} + 2 n x - 3 {n}^{2} = 0$

Explanation:

Let:

$f \left(x\right) = \left(x - n\right) \left(x + 3 n\right) = {x}^{2} + 2 n x - 3 {n}^{2}$

Then $f \left(x\right) = 0$ if and only if at least one of $\left(x - n\right) = 0$ or $\left(x + 3 n\right) = 0$

That is, if and only if $x = n$ or $x = - 3 n$

So a suitable quadratic equation would be:

${x}^{2} + 2 n x - 3 {n}^{2} = 0$

Any quadratic equation in $x$ with these zeros is a non-zero scalar multiple of this equation.