Problem 12 - Elementary Algebra

Question

For the following problems, solve the square root equations. $$ \sqrt{x}=1 $$

Step-by-Step Solution

Answer

Answer: The value of x that satisfies the equation is x=1.

Step 1: Analyze the equation

We have the equation: $$ \sqrt{x}=1 $$ Our task is to find the value of x that satisfies this equation.

Step 2: Square both sides of the equation

In order to eliminate the square root, we need to square both sides of the equation. In this case, we have: $$ (\sqrt{x})^2 = (1)^2 $$

Step 3: Solve for x

After squaring both sides, the equation becomes: $$ x = 1 $$ In this case, we can see that the value of x that satisfies the equation is x = 1.

Key Concepts

Algebraic Solutions

Algebraic solutions refer to finding the value or set of values that satisfy an equation by manipulating algebraic expressions. To do so, the student must understand and apply various algebraic axioms and be comfortable with different operations.

In the context of solving square root equations, algebraic solutions involve clearing the radical sign to isolate the variable of interest. The process often includes operations such as squaring both sides to eliminate square roots, rearranging the equation, and simplifying terms. For the equation \(\sqrt{x} = 1\), we're in the pursuit of an algebraic solution that provides the value for x. The solution involves isolating x and making it the subject of the formula, hence the progression from the given equation to \(x = 1\) after the appropriate operations have been performed.

Squaring Both Sides

The technique of squaring both sides of an equation is a pivotal step when you are dealing with square root equations. The principle behind it is to apply the same operation to both sides of the equation to maintain equality. Squaring is particularly useful because it eliminates the square root sign, allowing us to get closer to finding the value of the variable.

It's important to remember that when you square both sides, you must square the entire side and not just the terms individually. For instance, when the equation \(\sqrt{x} = 1\) is given, by squaring both the expression containing the root and the constant (1), you get \(\sqrt{x}^2 = 1^2\), which then simplifies to \(x = 1\). This technique transforms the equation into a simpler form that we can solve readily.

Root Extraction

Root extraction is the process of 'undoing' a square root within an equation. It's a reverse operation to squaring and is often used as a verification step after squares have canceled out on both sides of an equation. In our textbook exercise, after squaring gets rid of the square root, we are effectively extracting the root.

This method simplifies the equation further to find the solution for x. The process may seem redundant in a direct equation like \(\sqrt{x} = 1\), where squaring both sides immediately gives us the answer. However, in more complex equations, after squaring, the solution for x, or whichever variable is in question, could still involve several steps of simplification and root extraction to arrive at the final answer.