Problem 392 - Elementary Algebra

Question

In the following exercises, check whether the given values are solutions. For the equation \(\sqrt{u+42}=u:\) (a) Is \(u=-6\) a solution? (b) Is \(u=7\) a solution?

Step-by-Step Solution

Answer

No, \(u = -6\) is not a solution. Yes, \(u = 7\) is a solution.

Step 1: Understand the equation

The equation given is \ \sqrt{u+42}=u. We need to check if the values of \(u\) provided (\(u = -6\) and \(u = 7\)) satisfy this equation.

Step 2: Check if \(u = -6\) is a solution

Substitute \(u = -6\) into the equation: \ \sqrt{-6+42} = -6. Simplify inside the square root: \ \sqrt{36}= -6. Since \( \sqrt{36} = 6\) and \(6 eq -6\), \(u = -6\) is not a solution.

Step 3: Check if \(u = 7\) is a solution

Substitute \(u = 7\) into the equation: \ \sqrt{7+42}=7. Simplify inside the square root: \ \sqrt{49}=7. Since \(\sqrt{49}=7\) and \(7=7\), \(u = 7\) is a solution.

Key Concepts

Verifying Solutions

In mathematics, verifying solutions is an essential step to ensure the correctness of your results. It involves substituting the given values back into the original equation and checking if the equation holds true. For example, if we have the equation \(\root{u+42}=u\), we can verify if a value such as \(u=-6\) or \(u=7\) is a solution by substituting these values back into the equation. Let's break it down further:
The step-by-step process involves:

• Identifying the original equation.
• Substituting the given value into the equation.
• Simplifying and checking if both sides of the equation are equal.

For instance, substituting \(u=-6\) into the equation \(\root{u+42}=u\):
\[\root{-6+42}=-6\]
This simplifies to:
\[\root{36}=-6\]
We know that \(\root{36}=6\), and since 6 is not equal to -6, \(-6\) is not a solution. Similarly, verifying \7\:
\[\root{7+42}=7\]
This simplifies to:
\[\root{49}=7\]
Since \(\root{49}=7\), both sides of the equation are equal, so \(u=7\) is indeed a solution.
Verifying solutions help confirm the correctness of our results.

Square Root Equations

Square root equations are equations in which a square root expression is set equal to another expression. They often appear in the form:

• \root{x}=b\
• \root{x+a}=b\

These kinds of equations require careful manipulation to solve because squaring both sides of the equation can introduce extraneous solutions—values that don't actually satisfy the original equation.
In the provided exercise, we have the equation:
\[\root{u+42}=u\]
To solve or verify solutions for such an equation, you need to follow these steps:

• Isolate the square root on one side, if it's not already isolated.
• Square both sides of the equation to eliminate the square root.
• Solve the resulting standard equation.

For example, let's take the given equation again:
\[\root{u+42}=u\]
To eliminate the square root, we square both sides:
\[ (\root{u+42}})^2 = u^2 \]
This simplifies to:
\[u + 42 = u^2 \]
Solving for \(u\) gives you potential solutions, which you can then verify. Always double-check your solutions by substituting them back into the original equation to ensure they do not introduce extraneous values.

Substitution Method

The substitution method is a technique used to solve equations, especially systems of equations. It involves substituting one equation into another to simplify the problem and find the variable's value. Here's how you can use this method:

• Substitute the given value into the equation.
• Simplify the equation by performing algebraic operations.
• Solve for the variable.

In our exercise, we use substitution to check if a given value is a solution. For instance, substituting \(u=7\) into the equation \(\root{u+42}=u\) looks like:
\[\root{7+42}=7\]
Simplifying inside the square root, we get:
\[\root{49}=7\]
As \(\root{49}=7\), this confirms \(u=7\) as a valid solution. This method is especially handy when you have values to verify. Simply substitute them and see if they satisfy the original equation. Essentially, substitution is about replacing and simplifying the equation to see if everything holds true.