Problem 50 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Isolate the square root

First, isolate the square root on one side of the equation by subtracting 2 from both sides. This yields: \(f(x) - 2 = \sqrt{x}\)

Step 2: Squaring both sides

To eliminate the square root, square both sides of the equation. This is done using the rule \((\sqrt[n]{y})^{n} = y\), where in this case \(n = 2\). The equation becomes: \((f(x) - 2)^2 = x \)

Step 3: Final form of the equation

The equation is now solved for \(x\). The final form of the equation is: \(x = (f(x) - 2)^2\).

Key Concepts

Isolate the Square Root

Solving equations involving square roots often begins with the critical step of isolating the square root. Think of it as giving the square root its own space on one side of the equation. This step is essential because it sets the stage for removing the square root altogether, paving the way to find the solution to the equation. For instance, consider an equation like
\( f(x) = \sqrt{x} + 2\).
To isolate the square root, we deduct 2 from both sides, which gives us
\( f(x) - 2 = \sqrt{x}\),
leaving the square root unaccompanied. This act simplifies the process that follows, which is elevating both sides to a power to eliminate the square root.

Raising Both Sides to a Power

Once the square root is on its own, the next step is to remove it by 'raising both sides to a power,' typically the power of 2, because \( (\sqrt[n]{y})^{n} = y\). This operation is akin to unlocking the square root to free the variable inside. For example, squaring the isolated square root from our previous step results in
\( (f(x) - 2)^2 = x\).
This manoeuvre effectively dispels the square root, translating our equation into a more familiar algebraic form that can finally be solved. It's crucial to proceed with caution here—squaring both sides introduces the possibility of extraneous solutions, which are solutions that might not satisfy the original equation. That's why a verification step is a must after obtaining potential solutions.

Square Root Functions

Understanding Square Root Functions

A square root function is a type of function that involves the square root of a variable. The general form of a square root function is
\( f(x) = \sqrt{x} \),
often accompanied by additions, subtractions, multiplications, or divisions. These functions are unique because they only output real numbers for non-negative input values, due to the fact that the square root of a negative number is not real in the realm of real numbers. When graphing square root functions, the result is a curve that starts at the origin (0,0) or is shifted according to any added constants. The curve typically extends only to the right since square roots of negative numbers are not considered when dealing with real-valued functions.

Algebraic Equations

Algebraic equations are the bread and butter of solving mathematical problems involving unknown variables. They consist of expressions set equal to each other, containing constants, variables, and operations such as addition, subtraction, multiplication, and division.
The art of solving these equations lies in the manipulation of these terms to isolate and solve for a variable. It's an intricate dance that often involves multiple steps, such as distributing, combining like terms, and using inverse operations to simplify the equation down to its most basic form where the variable stands alone. In the case of square root equations, after removing the square roots and isolating the variable, it’s imperative to check all potential solutions back in the original equation to ensure they are valid, as some operations can introduce false leads that don’t actually solve the initial problem.