We are given the equation:

\[
x^2 - \frac{1}{x^2} = 12
\]

Let's solve this step by step.

### Step 1: Introduce a substitution
Let:

\[
y = x^2 + \frac{1}{x^2}
\]

Then, square both sides of this new equation:

\[
y^2 = \left(x^2 + \frac{1}{x^2}\right)^2
\]

Using the identity \((a + b)^2 = a^2 + 2ab + b^2\), we get:

\[
y^2 = x^4 + 2 + \frac{1}{x^4}
\]

Thus:

\[
y^2 - 2 = x^4 + \frac{1}{x^4}
\]

Now, return to our original equation:

\[
x^2 - \frac{1}{x^2} = 12
\]

Let \( z = x^2 - \frac{1}{x^2} \

Step-by-step solution to the algebraic problem x^2 - 1/x^2 = 12 using substitution and identities. This solution covers key algebraic concepts like rational expressions and the square of a sum. Suitable for students in Grades 10-12.

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