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Chapter 8: Problem 26

Solve each equation. $$ \sqrt{x+2}=\sqrt{2 x-5} $$

Short Answer

Expert verified
The solution is \(x = 7\).

Step by step solution

01

Eliminate the Square Roots

Square both sides of the equation to eliminate the square roots. \[ (\sqrt{x+2})^{2} = (\sqrt{2x-5})^{2} \]This results in the equation: \[ x+2 = 2x-5 \]
02

Isolate the Variable

Subtract \(x\) from both sides to get all the \(x\)-terms on one side.\[ x + 2 - x = 2x - 5 - x \]This simplifies to:\[ 2 = x - 5 \]
03

Solve for \(x\)

Add 5 to both sides of the equation to solve for \(x\).\[ 2 + 5 = x - 5 + 5 \]This simplifies to:\[ x = 7 \]
04

Check the Solution

Substitute \(x = 7\) back into the original equation to verify the solution.\[\sqrt{7 + 2} = \sqrt{2(7) - 5} \]Simplify both sides:\[\sqrt{9} = \sqrt{14 - 5} \]\[ 3 = 3 \]Since both sides are equal, \(x = 7\) is the correct solution.

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Key Concepts

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

Eliminating Square Roots
When solving radical equations, one of the first tasks is to eliminate the square roots. This can be done by squaring both sides of the equation. In the given exercise, we start with: \(\sqrt{x+2}=\sqrt{2x-5}\). By squaring both sides we use the property that \((\sqrt{a})^{2}=a\) for any non-negative \(a\). Therefore, squaring both sides we get:
\[(\sqrt{x+2})^{2} =(\sqrt{2x-5})^{2}\]
This effectively removes the square roots, resulting in the simpler equation: \(x+2 = 2x-5\). Now, we have a basic linear equation. Taking out the square roots makes the equation easier to work with.
Isolating Variables
To solve for the variable \(x\), we need to get all the \(x\) terms on one side and constants on the other. Starting from: \(x+2 = 2x-5\)
1. Subtract \(x\) from both sides:
\(x + 2 - x = 2x - 5 - x\).
2. This simplifies to: \(2 = x - 5\).

In this step, we successfully isolate \(x\) to one side. Isolating variables often involves adding, subtracting, multiplying, or dividing all parts of an equation.
Checking Solutions
Verifying our solutions is crucial. We substitute \(x = 7\) back into the original equation:
\(\sqrt{7 + 2} = \sqrt{2(7) - 5}\).
1. Simplify inside the square roots:
\(\sqrt{9}= \sqrt{14 - 5}\).
2. This gives us: \(\sqrt{9}= \sqrt{9}\).
3. The square root of 9 is 3, so we have: \(3 = 3\).

Because both sides of the equation are equal, our solution \(x = 7\) is verified. Always check solutions to ensure no mistakes were made during solving.
Simplifying Equations
Simplifying equations means reducing them to their most basic form. In our problem, after eliminating the square roots and isolating the variable, we simplified:
1. From \(x+2 = 2x-5\)
2. To \(2 = x - 5\)
3. Finally, to \(x = 7\).

This simplification process involves performing basic arithmetic operations like addition, subtraction, multiplication, and division. Breaking down complex equations step-by-step helps make them easier to solve.

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