Question

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

Answer 1

`a = pm 225`

Explanation:

We have: `x^(2) = a`

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

`Rightarrow x^(2) - a = 0`

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

`Rightarrow (x - sqrt(a))(x + sqrt(a)) = 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`:

`Rightarrow sqrt(a) - (- sqrt(a)) = 30 Rightarrow sqrt(a) + sqrt(a) = 30 Rightarrow 2 sqrt(a) = 30 Rightarrow sqrt(a) = 15 therefore a = 225`

`or`

`Rightarrow - sqrt(a) - sqrt(a) = 30 Rightarrow - 2 sqrt(a) = 30 Rightarrow sqrt(a) = - 15 therefore a = - 225`

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