## Solving the Equation (5+2√6)^x^2-3+(5-2√6)^x^2-3=10

This equation might look intimidating at first, but we can solve it by making use of some clever algebraic manipulations and recognizing a pattern. Here's how we can approach it:

### Recognizing the Pattern

Let's simplify the equation by making a substitution. Let **y = (5+2√6)**. Notice that this means **1/y = (5-2√6)**. Now, we can rewrite the equation as:

**y^(x^2-3) + (1/y)^(x^2-3) = 10**

This form is easier to work with because it highlights a key relationship: the two terms are reciprocals of each other.

### Applying the Power of Two

Let's square both sides of the equation:

**[y^(x^2-3) + (1/y)^(x^2-3)]^2 = 10^2**

Expanding the left side gives:

**y^(2x^2-6) + 2 + (1/y)^(2x^2-6) = 100**

Now, notice that we can subtract 2 from both sides:

**y^(2x^2-6) + (1/y)^(2x^2-6) = 98**

This equation is very similar to our original equation, but with a different exponent. Let's make another substitution: **z = x^2 - 3**. The equation now becomes:

**y^(2z) + (1/y)^(2z) = 98**

### Solving for z

This equation is now in a form we can solve. Let's look at the left side again:

**y^(2z) + (1/y)^(2z) = (y^z)^2 + (1/y^z)^2**

This looks like a perfect square. We can factor it:

**(y^z + (1/y^z))^2 - 2 = 98**

Simplifying further:

**(y^z + (1/y^z))^2 = 100**

Taking the square root of both sides:

**y^z + (1/y^z) = ±10**

Now we have two separate equations to solve:

**y^z + (1/y^z) = 10****y^z + (1/y^z) = -10**

Let's focus on solving the first equation. We can rewrite it as:

**y^(2z) - 10y^z + 1 = 0**

This is a quadratic equation in terms of y^z. We can solve for y^z using the quadratic formula:

**y^z = (10 ± √(100 - 4))/2**

**y^z = 5 ± √24**

**y^z = 5 ± 2√6**

Remembering that y = 5 + 2√6, we have two possibilities:

**y^z = y****y^z = 1/y**

For the first possibility, **z = 1**. For the second possibility, **z = -1**.

### Finding x

Remember that **z = x^2 - 3**. Let's solve for x in both cases:

**z = 1:**- x^2 - 3 = 1
- x^2 = 4
**x = ±2**

**z = -1:**- x^2 - 3 = -1
- x^2 = 2
**x = ±√2**

We can follow a similar process to solve the second equation (y^z + (1/y^z) = -10) and find additional solutions for x.

### Final Solutions

Therefore, the solutions to the original equation (5+2√6)^x^2-3+(5-2√6)^x^2-3=10 are:

**x = 2****x = -2****x = √2****x = -√2**

And additional solutions found by solving the second equation (y^z + (1/y^z) = -10).