Problem 67 - Algebra and Trigonometry

Question

The given equation involves a power of the variable. Find all real solutions of the equation. \((x+2)^{4}-81=0\)

Step-by-Step Solution

Answer

The real solutions are \(x = 1\) and \(x = -5\).

Step 1: Isolate the Power Expression

Start by adding 81 to both sides of the equation to isolate the power expression. This gives us \((x+2)^4 = 81\).

Step 2: Take the Fourth Root of Both Sides

To solve for \(x\), take the fourth root of both sides. This gives us \(x+2 = \pm \sqrt[4]{81}\). Since \(81\) is a perfect fourth power, \(\sqrt[4]{81} = 3\). So, \(x+2 = \pm 3\).

Step 3: Solve for x

Now, solve for \(x\) by setting up two separate equations: \(x+2 = 3\) and \(x+2 = -3\). Solve these equations individually.

Step 4: Solve the First Equation

For \(x+2 = 3\), subtract 2 from both sides to find \(x = 1\).

Step 5: Solve the Second Equation

For \(x+2 = -3\), subtract 2 from both sides to find \(x = -5\).

Step 6: List All Real Solutions

The solutions to the equation are \(x = 1\) and \(x = -5\). Both satisfy the original equation.

Key Concepts

Real Solutions

In the context of polynomial equations, determining real solutions involves finding values that satisfy the equation perfectly when substituted back into the original form. Real solutions are numbers that do not have imaginary components, meaning they can be plotted along the real number line.
For example, when solving the equation \((x+2)^{4}-81=0\), the values of \(x\) that make the equation true are called real solutions. Real solutions are important because they represent actual, possible values in practical situations, unlike complex solutions which arise when dealing with square roots of negative numbers.
In this exercise, our real solutions were discovered during the solve for x steps, resulting in \(x = 1\) and \(x = -5\). These numbers satisfy the equation when substituted back.

Fourth Root

To solve equations involving power expressions, such as \((x+2)^4 = 81\), taking the root relative to the power is essential. In this case, we take the fourth root, as the expression is raised to the fourth power.
Taking the fourth root of both sides simplifies the equation significantly: \(x+2 = \pm \sqrt[4]{81}\).

• The positive fourth root of 81 is 3, since \(3^4 = 81\).
• The negative fourth root is -3, yielding the value on the negative side, -3.

The fourth root helps reduce complexity by "undoing" the fourth power. This step is crucial to moving from a power expression to simpler, solvable equations.

Isolate Power Expression

Isolating the power expression means rewriting the equation so that the term with the power is by itself on one side. For \((x+2)^{4} - 81 = 0\), this step involves basic algebraic manipulation. To isolate, add 81 to both sides to get \((x+2)^4 = 81\).

• This single-step adjustment prepares the base of the expression for further manipulation like taking roots.
• In any similar equation, isolating the power term is the first step to simplifying the solution process.

By isolating the power expression, you're effectively setting the stage to solve the equation more straightforwardly.

Perfect Power

A perfect power is a number that can be expressed as another number raised to an integer power. For instance, 81 is a perfect fourth power because it is equal to \(3^4\). Recognizing perfect powers in equations allows for straightforward simplification.
By knowing \(81 = 3^4\), you can easily compute roots, such as the fourth root in this problem, leading to logical and clear-cut solutions.

• This knowledge breaks down complex expressions into manageable parts.
• Recognizing perfect squares, cubes, or fourth powers spares one from unnecessary calculation.

So when finding the fourth root of a perfect fourth power like 81, you simply calculate \(x+2 = 3\) or \(x+2 = -3\), directly arriving at real solutions.