(5+2√6)^x^2-3+(5-2√6)^x^2-3=10

Jasonbradley

5 min read

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Jun 16, 2024

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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:

1. y^z + (1/y^z) = 10
2. 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:

1. y^z = y
2. 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:

1. z = 1:
   • x^2 - 3 = 1
   • x^2 = 4
   • x = ±2
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).