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Chapter 10: Problem 21

Solve each equation. $$ 3 \sqrt{x-1}=2 \sqrt{2 x+2} $$

Short Answer

Expert verified
The solution is verified to be x=3].

Step by step solution

01

Isolate one of the square roots

Choose one of the square roots to isolate. For this problem, start by dividing both sides by 2.
02

Square both sides

To eliminate the square roots, square both sides of the equation.
03

Simplify the equations

After squaring both sides, simplify the resulting equation:
04

Solve the resulting equation

Solve the simplified equation for the variable x.
05

Check the solution

Verify the solution by substituting it back into the original equation to make sure it satisfies the equation.

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Key Concepts

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

isolating square roots
When solving square root equations, the first step is to isolate one of the square roots on one side of the equation. This helps in simplifying the equation later on.
If the equation looks like this:
\( 3 \sqrt{x-1}=2 \sqrt{2x+2} \)
We can start by dividing both sides by 2 or 3 to isolate a square root.
Let's divide both sides by 3:
\( \sqrt{x-1} = \frac{2} {3} \sqrt{2x+2} \)
Now, one of the square roots is isolated and we can proceed to the next step.
squaring both sides
Once we have isolated one of the square roots, the next step is to eliminate the square roots by squaring both sides of the equation.
This step is crucial because it allows us to work with simpler algebraic expressions.
If we go back to our isolated equation:
\( \sqrt{x-1} = \frac{2} {3} \sqrt{2x+2} \)
Squaring both sides:
\( ( \sqrt{x-1} )^{2} = ( \frac{2} {3} \sqrt{2x+2} )^{2} \)
This simplifies to:
\( x - 1 = \frac{4} {9} ( 2x + 2 ) \)
simplifying equations
After squaring both sides, we need to simplify the resulting equation to solve for the unknown variable.
Starting with the equation:
\( x - 1 = \frac{4} {9} ( 2x + 2 ) \)
Expand the right-hand side:
\( x - 1 = \frac{8x + 8} {9} \)
To clear the fraction, multiply both sides by 9:
\( 9(x - 1) = 8x + 8 \)
Distribute the 9:
\( 9x - 9 = 8x + 8 \)
Subtract 8x from both sides:
\( x - 9 = 8 \)
Finally, add 9 to both sides:
\( x = 17 \)
verifying solutions
After solving for the variable, it's important to verify the solution by substituting it back into the original equation to ensure it satisfies the equation.
Let's substitute \( x = 17 \) into the original equation:
\( 3 \sqrt{17-1} = 2 \sqrt{2(17)+2} \)
Simplify inside the square roots:
\( 3 \sqrt{16} = 2 \sqrt{34 + 2} \)
Further simplifying:
\( 3 (4) = 2 \sqrt{36} \)
\( 12 = 2 (6) \)
\( 12 = 12 \)
The solution satisfies the original equation, so \( x = 17 \) is correct.

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