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Chapter 9: Problem 418

In the following exercises, solve. $$ \sqrt{8+2 r}=\sqrt{3 r+10} $$

Short Answer

Expert verified
The value of r is -2.

Step by step solution

01

Square Both Sides

To eliminate the square roots, square both sides of the equation. \( (\text{sqrt}(8 + 2r))^2 = (\text{sqrt}(3r + 10))^2 \rightarrow 8 + 2r = 3r + 10 \)
02

Simplify the Equation

Rearrange the equation to isolate the variable term. Subtract 2r from both sides to collect all the terms involving r on one side.\(8 + 2r - 2r = 3r + 10 - 2r \rightarrow 8 = r + 10\)
03

Solve for r

Subtract 10 from both sides to isolate r.\(8 - 10 = r + 10 - 10 \rightarrow -2 = r \)
04

Verify the Solution

Plug the value of r back into the original equation to ensure it satisfies both sides. \(\text{sqrt}(8 + 2(-2)) = \text{sqrt}(3(-2) + 10)\) Calculate both sides separately. \(\text{sqrt}(8 - 4) = \text{sqrt}(4) \) \(\text{sqrt}(4) = 2 \text{ and } \text{sqrt}(4) = 2 \) Both sides are equal, confirming that the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Squaring Both Sides
When solving a radical equation, removing the square root is often the first step. The way to get rid of the square root is by squaring both sides of the equation. This is because the square of a square root returns the original value under the root. For example, squaring both sides of \( \sqrt{8 + 2r} = \sqrt{3r + 10} \) gives: \( (\sqrt{8 + 2r})^2 = (\sqrt{3r + 10})^2 \). Simplifying this, we get: \( 8 + 2r = 3r + 10 \). Ensure to square correctly to avoid errors in subsequent steps.
Isolating Variable
After squaring both sides, you need to isolate the variable. In this problem, the squared equation is \( 8 + 2r = 3r + 10 \). To isolate the variable \( r \), you'll want the terms involving \( r \) on one side and the constants on the other. Start by subtracting \( 2r \) from both sides: \( 8 + 2r - 2r = 3r + 10 - 2r \). This simplifies to: \( 8 = r + 10 \).
Verifying Solution
Verification is crucial in solving equations because sometimes squaring can introduce extraneous solutions, which don't satisfy the original equation. Substituting \( r = -2 \) back into the original equation: \( \sqrt{8 + 2(-2)} = \sqrt{3(-2) + 10} \). Calculating both sides: \( \sqrt{8 - 4} = \sqrt{4} \) and \( \sqrt{4} = 2 \). Since both sides equal, the solution \( r = -2 \) is verified.
Simplifying Equations
Simplifying equations is essential in the process of solving. This involves rearranging terms to make them easier to work with. For instance, from \( 8 = r + 10 \), you subtract 10 from both sides: \( 8 - 10 = r + 10 - 10 \). This simplifies to: \( -2 = r \). Keep the equations as simple as possible to reduce the risk of errors and make further steps straightforward.

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