How do you simplify (sqrt5-x)(sqrt5+x)?

Oct 4, 2017

Use the FOIL method of multiplication, and the result is $5 - {x}^{2}$

Explanation:

The FOIL method, usually thought of as applying to binomial expressions can be used for this radical expression as well.

It tells us to multiply the First terms in each bracket, the Outside terms, the Inside terms and the Last terms.

Thus,

$\left(\sqrt{5} - x\right) \cdot \left(\sqrt{5} + x\right) = \left[\left(\sqrt{5} \cdot \sqrt{5}\right) + \sqrt{5} \cdot x + \sqrt{5} \cdot \left(- x\right) + x \cdot \left(- x\right)\right]$

Simplifying:

$\left(\sqrt{5} - x\right) \cdot \left(\sqrt{5} + x\right) = \left(5 + \sqrt{5} \cdot x - \sqrt{5} \cdot x - {x}^{2}\right)$

Noting that the second and third terms add to zero, we get the final result:

$\left(\sqrt{5} - x\right) \cdot \left(\sqrt{5} + x\right) = 5 - {x}^{2}$