/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 42 \(\sqrt{r+6}-\sqrt{r-2}=2\)... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 42

\(\sqrt{r+6}-\sqrt{r-2}=2\)

Short Answer

Expert verified
The value of \( r \) is 3.

Step by step solution

01

- Isolate one of the square roots

Move one of the square root terms to the other side of the equation. In this case, isolate \ \(\sqrt{r+6} \): \ \[\sqrt{r+6} = \sqrt{r-2} + 2\]
02

- Square both sides

To eliminate the square roots, square both sides of the equation: \ \[(\sqrt{r+6})^2 = (\sqrt{r-2} + 2)^2 \] Simplify both sides: \ \[r+6 = (\sqrt{r-2})^2 + 4\sqrt{r-2} + 4 \]\[r+6 = r-2 + 4\sqrt{r-2} + 4 \]
03

- Simplify

Combine like terms on the right side of the equation: \ \[r+6 = r + 2 + 4\sqrt{r-2} \]Next, isolate the term with the square root: \ \[6 = 2 + 4\sqrt{r-2} \] \[4 = 4\sqrt{r-2} \] Divide both sides by 4: \ \[1 = \sqrt{r-2} \]
04

- Square both sides again

Square both sides again to eliminate the square root: \ \[(1)^2 = (\sqrt{r-2})^2 \] Simplify: \ \[1 = r-2 \]
05

- Solve for r

Add 2 to both sides to solve for \r\: \[1 + 2 = r \] \[r = 3 \]
06

- Verify the solution

Substitute \r = 3\ back into the original equation to ensure it works: \[\sqrt{3+6} - \sqrt{3-2} = 2 \sqrt{9} - \sqrt{1} = 2 \3 - 1 = 2 \2 = 2 \] The solution is verified.

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Key Concepts

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

Isolating Terms
The first step in solving equations involving square roots is to isolate one of the terms with a square root. This simply means moving one of the square root expressions to one side of the equation. For instance, consider the equation \(\root{r+6} - \root{r-2} = 2\). To isolate \(\root{r+6}\), we add \(\root{r-2}\) to both sides of the equation: \(\root{r+6} = \root{r-2} + 2\).
• This rearranges the equation and sets the stage for the next steps.
• Remember, isolating terms helps in dealing with one unknown at a time.

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