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Chapter 8: Problem 22

(a) solve. (b) check. $$\sqrt{2 x+8}=12$$

Short Answer

Expert verified
x = 68

Step by step solution

01

Isolate the Square Root

The given equation is \(\sqrt{2x + 8} = 12\). Since the square root is already isolated, move on to the next step.
02

Square Both Sides

Square both sides of the equation to eliminate the square root:\((\sqrt{2x + 8})^2 = 12^2\).This simplifies to:\(2x + 8 = 144\).
03

Solve for x

Now solve the equation by isolating \(x\).Subtract 8 from both sides:\(2x + 8 - 8 = 144 - 8\)\(2x = 136\).Then divide both sides by 2:\(x = 68\).
04

Check the Solution

Substitute \(x = 68\) back into the original equation to verify the solution:\(\sqrt{2(68) + 8} = \sqrt{136 + 8} = \sqrt{144}\).Since \(\sqrt{144} = 12\), the original equation holds true.

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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 equations involving square roots, the first important step is to isolate the square root expression. This means we want to get the square root by itself on one side of the equation.

In our example equation \(\sqrt{2x + 8} = 12\)\, the square root is already by itself on the left side. Hence, no further isolation is needed. However, in other equations, you might need to perform some operations such as:
  • Adding or subtracting terms from both sides
  • Dividing or multiplying both sides by a number
These steps help in achieving the square root alone on one side of the equation, making the next steps easier.
Squaring Both Sides
Once the square root is isolated, the next step is to square both sides of the equation to get rid of the square root. Squaring is when we multiply a number by itself. For example, \(3^2 = 3 \times 3 = 9\). By squaring both sides of \(\sqrt{2x+8} = 12\)we have:
  • \(\sqrt{2x+8}^2 = 12^2\)
Squaring the square root on the left side simplifies it to \(2x + 8\), since taking the square root and then squaring returns the initial value under the square root.
The right side becomes: \(12^2 = 144\).

Now we have a simpler equation: \(2x + 8 = 144\). This is much easier to solve.
Checking Solutions
The last step in solving square root equations is to check if the potential solution satisfies the original equation. This step is crucial because squaring both sides of an equation can sometimes introduce extraneous solutions (solutions that do not actually satisfy the original equation).
To check our solution, we substitute \(x = 68\) back into the original equation:

Substitute \(x = 68\) into \(\sqrt{2x +8}\):
  • \(\sqrt{2(68) + 8} = \sqrt{136 + 8} = \sqrt{144} = 12\)
Since both sides match, \(x = 68\) is indeed a correct solution.
Always ensure your final answers satisfy the original equation to confirm their validity.

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