Question

Find the quadratic equation whose roots are:

$2\sqrt{2}$ and $-3\sqrt{2}$

Answer 1

Step 1: Given Information and use the relation of the roots of a quadratic equation.

The roots of the quadratic equation are $2\sqrt{2}$ and $-3\sqrt{2}$.

We know that the sum of roots is:

$\begin{array}{rcl}& \Rightarrow & \alpha +\beta =2\sqrt{2}-3\sqrt{2}\\ & =& -1\sqrt{2}\\ & =& -\sqrt{2}\end{array}$

We know that the product of roots is:

$\begin{array}{rcl}& \Rightarrow & \alpha \times \beta =2\sqrt{2}\times -3\sqrt{2}\\ & =& -12\end{array}$

Step 2: Obtain the equation.

The general quadratic equation is in the form of:

$x^2-\left(\alpha +\beta \right)x+\left(\alpha ·\beta \right)=0$

Substitute obtained values to get:

$\begin{array}{rcl}{x}^2-\left(\alpha +\beta \right)x+\left(\alpha ·\beta \right)& =& 0\\ x^2+\sqrt{2}x-12& =& 0\end{array}$

Hence, the required equation is $x^2+\sqrt{2}x-12=0$.