Question

Solve each equation. $$ 3^{x^{2}-7}=27^{2 x} $$

Short answer

The solutions are \(x = 7\) and \(x = -1\).

Step-by-step solution

Express both sides with the same base

Rewrite both sides of the equation using the same base. Notice that 27 can be written as a power of 3. Since 27 is equal to 3^3, rewrite the equation as: \(3^{x^2 - 7} = (3^3)^{2x}\)

Simplify the right side

Simplify \((3^3)^{2x}\) by applying the power of a power property \((a^m)^n = a^{mn}\). This gives: \(3^{x^2 - 7} = 3^{6x}\)

Set the exponents equal to each other

Since the bases are the same, we can set the exponents equal to each other: \(x^2 - 7 = 6x\)

Rearrange into a quadratic equation

Rearrange the equation into standard quadratic form: \(x^2 - 6x - 7 = 0\)

Factor the quadratic equation

Factor the quadratic equation \(x^2 - 6x - 7\) into: \((x-7)(x+1) = 0\)

Solve for x

Solve for \(x\) by setting each factor to zero: \(x - 7 = 0\) or \(x + 1 = 0\). Thus, \(x = 7\) or \(x = -1\)

Key concepts

exponential functions

Exponential functions play a critical role in many mathematical problems, particularly in growth and decay models. An exponential function follows the form \(a^x\), where \(a\) is a constant base and \(x\) is the exponent. In our exercise, the equation \(3^{x^2 - 7} = 27^{2x}\) involves exponential functions on both sides.

The trick to solving such equations often involves rewriting each side using the same base. In this case, we know that 27 can be expressed as \(3^3\). This allows us to convert the equation to a common base, making it easier to handle the exponents.

Understanding how to manipulate exponential functions is fundamental in algebra. These skills extend beyond homework problems and are useful in science and engineering.

quadratic equations

Quadratic equations can be written in the standard form \(ax^2 + bx + c = 0\). They are called 'quadratic' because of the \(x^2\) term, which represents the square of the variable. In our exercise, after simplifying the exponential expression, we end up with a quadratic equation: \(x^2 - 6x - 7 = 0\).

Solve this type of equation by either factoring, completing the square, or using the quadratic formula \([x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}]\). For this exercise, the equation \(x^2 - 6x - 7 = 0\) is easily factorable.

Quadratic equations appear in many contexts, such as projectile motion and area problems. Mastering them gives you powerful tools to solve a wide variety of problems.

factoring

Factoring is a method used to solve quadratic equations and simplify algebraic expressions. It involves writing a polynomial as a product of its factors. For example, the quadratic equation \(x^2 - 6x - 7 = 0\) can be factored as \((x - 7)(x + 1) = 0\).

To factor quadratics, look for two numbers that multiply to give the constant term \(c\) and add to give the coefficient \(b\) of the linear term. Here, -7 and +1 multiply to -7 and add to -6, fitting perfectly.

Factoring can also be used to simplify more complex expressions and solve higher-degree polynomials. Practice makes perfect, and familiarity with different factoring techniques is essential for tackling various algebraic problems effectively.