Problem 402
Question
In the following exercises, solve. \(\sqrt{4 m+2}+2=6\)
Step-by-Step Solution
Verified Answer
\( m = \frac{7}{2} \)
1Step 1: Isolate the Square Root
Subtract 2 from both sides of the equation: \(\sqrt{4m + 2} + 2 - 2 = 6 - 2\) This simplifies to: \(\sqrt{4m + 2} = 4\)
2Step 2: Square Both Sides
Square both sides to eliminate the square root: \( (\sqrt{4m + 2})^2 = 4^2 \) This simplifies to: \( 4m + 2 = 16 \)
3Step 3: Solve for m
Subtract 2 from both sides to isolate the term with the variable: \( 4m + 2 - 2 = 16 - 2 \) This simplifies to: \( 4m = 14 \) Divide both sides by 4 to solve for m: \( m = \frac{14}{4} = \frac{7}{2} \)
Key Concepts
Isolating the square rootSquaring both sidesSimple algebraic equations
Isolating the square root
When you're dealing with an equation that involves a square root, the first goal is to isolate the square root on one side of the equation. This helps in simplifying the equation by removing extra constants or terms.
To illustrate: In this example, the initial equation is \( \sqrt{4m + 2} + 2 = 6 \).
Our aim is to isolate \( \sqrt{4m + 2} \). We do this by subtracting 2 from both sides:
\[ \sqrt{4m + 2} + 2 - 2 = 6 - 2 \]
The equation now simplifies to:
\[ \sqrt{4m + 2} = 4 \]
By isolating the square root, we now have an equation that is simpler to handle. Removing constants in this manner makes the next steps easier and more intuitive.
To illustrate: In this example, the initial equation is \( \sqrt{4m + 2} + 2 = 6 \).
Our aim is to isolate \( \sqrt{4m + 2} \). We do this by subtracting 2 from both sides:
\[ \sqrt{4m + 2} + 2 - 2 = 6 - 2 \]
The equation now simplifies to:
\[ \sqrt{4m + 2} = 4 \]
By isolating the square root, we now have an equation that is simpler to handle. Removing constants in this manner makes the next steps easier and more intuitive.
Squaring both sides
The next key step is to eliminate the square root by squaring both sides of the equation. This helps transform our current equation into a simpler algebraic form.
From the previous isolated square root, we have:
\[ \sqrt{4m + 2} = 4 \]
To remove the square root, square both sides:
\[ (\sqrt{4m + 2})^2 = 4^2 \]
On the left side, the square root and the square cancel each other out, leaving:
\[ 4m + 2 = 16 \]
By squaring both sides, the equation has become a regular linear equation, making it easier to continue simplifying.
This technique is essential in solving square root equations because it transforms them into more familiar forms.
From the previous isolated square root, we have:
\[ \sqrt{4m + 2} = 4 \]
To remove the square root, square both sides:
\[ (\sqrt{4m + 2})^2 = 4^2 \]
On the left side, the square root and the square cancel each other out, leaving:
\[ 4m + 2 = 16 \]
By squaring both sides, the equation has become a regular linear equation, making it easier to continue simplifying.
This technique is essential in solving square root equations because it transforms them into more familiar forms.
Simple algebraic equations
Once you have a simple algebraic equation, the remaining steps involve basic algebra to solve for the variable.
In our example, after squaring both sides, we have the equation:
\[ 4m + 2 = 16 \]
Our goal is to isolate the term with the variable \( m \). Start by subtracting 2 from both sides:
\[ 4m + 2 - 2 = 16 - 2 \]
Which simplifies to:
\[ 4m = 14 \]
Now, divide both sides by 4 to solve for \( m \):
\[ m = \frac{14}{4} = \frac{7}{2} \]
This final step of basic algebraic manipulation solves for the variable. By isolating the square root and then transforming the equation, we made it straightforward to apply these simple algebraic operations.
Remember, practicing these steps helps to reinforce the logical sequence and methods necessary for solving similar problems in the future.
In our example, after squaring both sides, we have the equation:
\[ 4m + 2 = 16 \]
Our goal is to isolate the term with the variable \( m \). Start by subtracting 2 from both sides:
\[ 4m + 2 - 2 = 16 - 2 \]
Which simplifies to:
\[ 4m = 14 \]
Now, divide both sides by 4 to solve for \( m \):
\[ m = \frac{14}{4} = \frac{7}{2} \]
This final step of basic algebraic manipulation solves for the variable. By isolating the square root and then transforming the equation, we made it straightforward to apply these simple algebraic operations.
Remember, practicing these steps helps to reinforce the logical sequence and methods necessary for solving similar problems in the future.
Other exercises in this chapter
Problem 400
In the following exercises, solve. \(\sqrt{4 u+2}-6=0\)
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In the following exercises, solve. \(\sqrt{5 q+3}-4=0\)
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In the following exercises, solve. \(\sqrt{6 n+1}+4=8\)
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In the following exercises, solve. \(\sqrt{2 u-3}+2=0\)
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