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Exponential equations are a type of algebraic equation where the variable appears in an exponent. Solving these equations often involves employing the property that if two expressions with the same base are equal, their exponents must also be equal. For instance, the given exercise

$$5^{x^{2}-12}=25^{2x}$$

, demands a transformation of both sides of the equation to have the same base. Once this is achieved, as in the provided solution, we can equate the exponents and proceed to solve for the variable.

This process of equating exponents is particularly helpful because it simplifies the complex structure of an exponential equation into a more familiar form. In the case of our exercise, this transformation results in a quadratic equation, which is easier to tackle with well-known methods, such as factoring or applying the quadratic formula.

Quadratic equations are fundamental in algebra and they are characterized by the highest degree of the variable being two, taking the standard form

$$ax^{2} + bx + c = 0$$.

The given exercise presents such a situation after comparing the powers of the exponential equation. This is evident from the rearrangement in Step 3 of the provided solution, resulting in the quadratic equation

$$x^{2}-4x-12=0$$.

Quadratic equations can be approached using several methods including factoring, completing the square, or using the quadratic formula. When factoring or completing the square is cumbersome or impossible, the quadratic formula is the most reliable tool to find the solutions for the variable.

The quadratic formula is a powerful tool for solving quadratic equations and can be represented as

$$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$.

It is derived from completing the square of the standard quadratic equation. The solution involves substituting the corresponding values of the coefficients 'a', 'b', and 'c' into the formula. In the given exercise, applying this formula to the equation

$$x^{2} - 4x - 12 = 0$$

leads to the discovery of two possible values for x, which are 6 and -2, as shown in the final step of the provided solution.

Understanding the quadratic formula's application is critical for students as it guarantees a method to solve any quadratic equation, regardless of whether the equation can be factored easily or not. This ensures that students can confidently find the roots of quadratic equations encountered in various mathematical and real-world problems.