# The difference between the solutions to the equation x^2 = a is 30. What is a?

Sep 21, 2017

$a = \pm 225$

#### Explanation:

We have: ${x}^{2} = a$

Let's subtract $a$ from both sides of the equation:

$R i g h t a r r o w {x}^{2} - a = 0$

Then, using the difference of two squares identity, we can solve for $x$:

$R i g h t a r r o w \left(x - \sqrt{a}\right) \left(x + \sqrt{a}\right) = 0$

$\therefore x = \pm \sqrt{a}$

Now, the difference between these two solutions of $x$ is $30$.

Let's use this fact to find the value of $a$:

$R i g h t a r r o w \sqrt{a} - \left(- \sqrt{a}\right) = 30 R i g h t a r r o w \sqrt{a} + \sqrt{a} = 30 R i g h t a r r o w 2 \sqrt{a} = 30 R i g h t a r r o w \sqrt{a} = 15 \therefore a = 225$

$\mathmr{and}$

$R i g h t a r r o w - \sqrt{a} - \sqrt{a} = 30 R i g h t a r r o w - 2 \sqrt{a} = 30 R i g h t a r r o w \sqrt{a} = - 15 \therefore a = - 225$

Therefore, the value of $a$ is either $225$ or $- 225$.