# If one of the root of the equation x^2+px+q=0 is the square of another then what is the relation between p and q?

Nov 12, 2017

See below.

#### Explanation:

We are told that ${x}^{2} + p x + q = 0$, and we are asked to find the relationship between $p$ and $q$. In order to approach this problem, you will need to know the following:

The square of the linear factor $\left(x + a\right)$ can be written as ${\left(x + a\right)}^{2}$ or ${x}^{2} + 2 a x + {a}^{2}$.

Based on this information, we know that $2 a = p$, and ${a}^{2} = q$. Therefore, $a = \sqrt{q}$. We can use substitution to find the relationship between $p$ and $q$:

$2 a = p$

$2 \left(\sqrt{q}\right) = p$

$2 \sqrt{q} = p$

$\left(\frac{p}{2}\right) = \sqrt{q}$

${\left(\frac{p}{2}\right)}^{2} = {\left(\sqrt{q}\right)}^{2}$

$q = {p}^{2} / 4$

I hope that helps!