Problem 50 - Precalculus Essentials

Question

To solve for a variable involving an nth root, raise both sides of the equation to the nth power: \(\left.(\sqrt[n]{y})^{n}=y .\right)\). $$f(x)=\sqrt{x}+2$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = (c-2)^2\).

Step 1: Write down the given equation

The given function is \(f(x)=\sqrt{x}+2\). Then let \(f(x)\) equal a constant value, say \(c\). So we can get \(c= \sqrt{x}+2\)

Step 2: Isolate the square root

Subtract 2 from both sides of the equation in order to isolate \(\sqrt{x}\) on the right side. This gives us: \(c-2 = \sqrt{x}\)

Step 3: Square both sides of the equation

Now, in order to eliminate the square root, square both sides of the equation. This results in: \((c-2)^2 = x\)

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra that involves finding the value(s) of a variable that makes a mathematical statement true. Typically, this process consists of a series of steps to simplify the equation and narrow down the possible values that satisfy it. When dealing with an equation, such as \(f(x) = \sqrt{x} + 2\), our objective is to isolate and solve for \(x\).
These problems often involve transforming the equation by performing equivalent operations on both sides. Depending on the equation's complexity, this might include addition, subtraction, multiplication, division, and sometimes more advanced operations like powers and roots.

Key steps to solving equations often include:

• Rearranging terms to establish an equation where the variable stands alone on one side.
• Utilizing inverse operations to isolate the variable, such as using subtraction to counteract addition.
• Ensuring the operations performed maintain the equality of the equation.

This process not only leads to finding the solutions (values of \(x\)) but also enhances problem-solving skills, which are crucial in mathematics. Understanding and applying these steps flexible enables tackling various algebraic challenges.

Isolation of Variables

Isolation of variables is an essential technique when solving algebraic equations. It refers to the process of manipulating an equation so that a particular variable is by itself on one side of the equation. This method is fundamental in identifying the value of the variable.

In the exercise, we begin with the equation \(c = \sqrt{x} + 2\). Our goal is to isolate \(x\). The first step involves moving all terms except \(\sqrt{x}\) to the other side of the equation:

• Subtract 2 from both sides to eliminate the additional constant alongside the square root.
• This results in \(c - 2 = \sqrt{x}\), where the term \(\sqrt{x}\) is now isolated.

By isolating the square root, we simplify the problem of solving for \(x\). This step is crucial as it reduces the complexity of the equation, making it easier to apply additional operations, such as squaring both sides, to find the variable's value. Isolation often involves using inverse operations strategically to remove all distractions from the variable in focus.

Square Root

The square root operation is a fundamental concept in mathematics. It involves finding a number which, when multiplied by itself, equals the given value. Symbolically, if \(y = x^2\), then \(x = \sqrt{y}\). In our task, we encountered the square root in the equation \(c = \sqrt{x} + 2\).

To resolve the equation and eliminate the square root, we square both sides of the equation. This powerful technique enables us to "cancel out" the square root, as shown in the step:

• From \(c - 2 = \sqrt{x}\), square both sides to yield \((c - 2)^2 = x\).
• Squaring both sides is effective because it neutralizes the square root, leaving us with a simpler equation.

Square roots often appear in problems involving quadratic equations or geometry, and understanding their manipulation is key to solving such problems. The reverse process, moving from a squared term to a square root, is also valuable in diverse mathematical contexts.