Animated Solution for Mathematics - Quadratic Equations: Solve for $x:(5+2\sqrt{6}{)}^{x^2-3}+(5-2\sqrt{6}{)}^{x^2-3}=10$

The equation $(5+2\sqrt{6}{)}^{x^2-3}+(5-2\sqrt{6}{)}^{x^2-3}=10$ appears daunting at first glance. However, in JEE Advanced mathematics, such problems often rely on identifying hidden symmetries within the algebraic structure.

Observe the bases: $(5+2\sqrt{6})$ and $(5-2\sqrt{6})$. These are conjugate irrational numbers. We calculate their product to reveal their relationship:

Since their product is $1$, we conclude that $(5-2\sqrt{6})=\frac{1}{5+2\sqrt{6}}$. The bases are reciprocals of each other, which is the critical insight needed to simplify the expression.

To simplify the landscape, we introduce a dummy variable $t$ to represent the first term:

Because the second base is the reciprocal of the first, the second term becomes:

The original exponential equation now transforms into a manageable algebraic form:

We clear the fraction by multiplying the entire equation by $t$, resulting in $t^2+1=10t$. Rearranging this into the standard quadratic form, we obtain:

Applying the quadratic formula $t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ with $a=1$, $b=-10$, and $c=1$:

We now return to our original variable $x$ by solving for the two possible values of $t$.

Case 1: $t=5+2\sqrt{6}$

Case 2: $t=5-2\sqrt{6}$

Since $5-2\sqrt{6}=(5+2\sqrt{6}{)}^{-1}$, we have:

The final solutions for the equation are $x\in \{2,-2,\sqrt{2},-\sqrt{2}\}$.