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Chapter 8: Problem 9

(a) solve. (b) check. $$\sqrt{x}+1=4$$

Short Answer

Expert verified
x = 9

Step by step solution

01

Isolate the Square Root

Subtract 1 from both sides of the equation to isolate the square root term: \[ \sqrt{x} + 1 - 1 = 4 - 1 \] which simplifies to: \[ \sqrt{x} = 3 \]
02

Square Both Sides

Square both sides of the equation to remove the square root: \[ (\sqrt{x})^2 = 3^2 \] which simplifies to: \[ x = 9 \]
03

Check the Solution

Substitute \(x = 9\) back into the original equation to verify: \[ \sqrt{9} + 1 = 4 \] which simplifies to: \[ 3 + 1 = 4 \] Since both sides of the equation are equal, the solution is correct.

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Key Concepts

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

Isolating Terms
When solving radical equations, the first step is often isolating the term that contains the radical. This means we want to get the square root expression by itself on one side of the equation. In the example provided, the equation is \(\sqrt{x} + 1 = 4\).
To isolate the square root term, we subtract 1 from both sides:
\[\sqrt{x} + 1 - 1 = 4 - 1\]
This simplifies to:
\[\sqrt{x} = 3\]
Now, the radical is by itself, and we can proceed to the next step.
  • Start by identifying the term with the radical.
  • Use basic arithmetic operations like addition or subtraction to isolate the term.
Remember, isolating the radical term makes it easier to apply further operations like squaring.
Squaring Equations
After isolating the radical term, we need to eliminate the square root to solve the equation. This is done by squaring both sides of the equation. In this case, we square both sides:
\((\sqrt{x})^2 = 3^2\)
This step removes the square root:
\[x = 9\]\
  • For any equation with \sqrt{a} = b\, squaring both sides turns it into \a = b^2\.
  • Always apply parentheses when squaring both sides to ensure clarity.
Squaring is a vital step that simplifies the equation, allowing us to find the value of the variable. However, be careful as squaring can sometimes introduce extraneous solutions that need to be checked.
Verification of Solutions
The final step after solving a radical equation is to verify the solution. This ensures that our solution is correct and that we haven't introduced any errors or extraneous solutions. To verify, substitute the found value back into the original equation. In our example, we substitute \x = 9\ back into the original equation:
\[\sqrt{9} + 1 = 4\]
This simplifies to:
\[3 + 1 = 4\]
Since both sides are equal, the solution is verified as correct.
  • Always substitute the solution back into the original equation.
  • Ensure both sides of the equation are equal to confirm the solution.
Verification is crucial as it rules out any potential mistakes made during isolation or squaring and confirms the accuracy of the result.

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Most popular questions from this chapter

Simplify. $$\frac{2 a^{\frac{3}{7}}}{3 a^{\frac{3}{7}}}$$

Rewrite the radical expression in exponential notation. $$\sqrt[4]{y^{3}}$$

Rewrite the radical expression in exponential notation. $$\sqrt[3]{x y^{2}}$$

Simplify. $$\sqrt{12}$$

Simplify. $$\sqrt[3]{-1}$$
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