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Chapter 8: Problem 10

(a) solve. (b) check. $$\sqrt{x}+1=5$$

Short Answer

Expert verified
The solution is \( x = 16 \). After checking, it is verified to be correct.

Step by step solution

01

Isolate the square root

Subtract 1 from both sides of the equation to isolate the square root term.\[ \sqrt{x} + 1 - 1 = 5 - 1 \implies \sqrt{x} = 4\]
02

Square both sides

Square both sides of the equation to eliminate the square root.\[ (\sqrt{x})^2 = 4^2 \implies x = 16 \]
03

Check the solution

Substitute \( x = 16 \) back into the original equation to verify:\[ \sqrt{16} + 1 = 5 \]\[ 4 + 1 = 5 \]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 the Square Root
To solve a square root equation, the first step is usually to isolate the square root on one side of the equation. Isolating the square root makes it easier to manipulate the equation and eventually eliminate the square root. For example, given the equation \( \sqrt{x}+1=5 \) You want the square root term \( \sqrt{x} \) by itself, with no additional terms around it. To achieve this, you subtract 1 from both sides of the equation:\[ \sqrt{x} + 1 - 1 = 5 - 1 \]This yields:\[ \sqrt{x} = 4 \]Now the square root term is isolated, and you're ready to move on to the next step.
Squaring Both Sides
Once the square root term is isolated, the next step is to square both sides of the equation. Doing this will eliminate the square root, allowing you to solve for the variable inside the square root. Here's how it's done in our example, where \( \sqrt{x} = 4 \):Square both sides:\[ (\sqrt{x})^2 = 4^2 \]This simplifies to:\[ x = 16 \]Squaring both sides works because squaring is the inverse operation of taking a square root. After squaring both sides, the equation becomes a simple one to solve.
Verifying Solutions
The final step in solving a square root equation is to verify the solution. Verifying ensures that the solution is correct and satisfies the original equation. Let鈥檚 check our solution for \( x=16 \). Substitute \( x \) back into the original equation \( \sqrt{x}+1=5 \):\[ \sqrt{16} + 1 = 5 \]Simplify the left side:\[ 4 + 1 = 5 \]Since both sides of the equation are equal, \( x=16 \) is a valid solution. It's important to always verify because squaring both sides can sometimes introduce extraneous solutions that don't satisfy the original equation.

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Most popular questions from this chapter

Simplify. $$\sqrt[3]{24}+\sqrt[3]{-81}$$

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