How do you use the difference of two squares formula to factor 2x^2 − 18?

May 6, 2015

You know that ${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$
$\left(2 {x}^{2} - 18\right) = \left(x \sqrt{2} + \sqrt{18}\right) \left(x \sqrt{2} - \sqrt{18}\right) =$
$= \left(x \sqrt{2} + \sqrt{9 \cdot 2}\right) \left(x \sqrt{2} - \sqrt{9 \cdot 2}\right) =$
$= \left(x \sqrt{2} + 3 \sqrt{2}\right) \left(x \sqrt{2} - 3 \sqrt{2}\right) =$
$= \sqrt{2} \left(x + 3\right) \sqrt{2} \left(x - 3\right) = 2 \left(x + 3\right) \left(x - 3\right)$

May 6, 2015

To factor using integers, we may first remove the common factor of $2$. That will leave a difference of perfect squares.

$2 {x}^{2} - 18 = 2 \left({x}^{2} - 9\right) = 2 \left({x}^{2} - {3}^{2}\right)$

$= 2 \left(x + 3\right) \left(x - 3\right)$