/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 \(2 \sqrt{x}=\sqrt{3 x+4}\)... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 19

\(2 \sqrt{x}=\sqrt{3 x+4}\)

Short Answer

Expert verified
x = 4

Step by step solution

01

- Square Both Sides

Square both sides of the given equation to eliminate the square roots. The equation is: \(2 \sqrt{x} = \sqrt{3 x + 4}\). When we square both sides, we get: \((2 \sqrt{x})^2 = (\sqrt{3x + 4})^2\). This simplifies to: \(4x = 3x + 4\).
02

- Simplify the Equation

Subtract \(3x\) from both sides to isolate \(x\) on one side of the equation: \(4x - 3x = 3x + 4 - 3x\). This simplifies to: \(x = 4\).
03

- Verify the Solution

Substitute \(x = 4\) back into the original equation to ensure it holds true: \(2 \sqrt{4} = \sqrt{3(4) + 4}\). This simplifies to: \(2 \cdot 2 = \sqrt{12 + 4}\) or \(4 = \sqrt{16}\). Since \(\sqrt{16} = 4\), both sides of the original equation are equal, verifying that the solution is correct.

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Key Concepts

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

Squaring Both Sides
To solve square root equations like the one given鈥擻(2 \sqrt{x} = \sqrt{3x + 4}\)鈥攚e often start by squaring both sides. Why square both sides? Squaring helps us get rid of the square roots, which makes the equation easier to solve.

Let's look at our equation:\(2 \sqrt{x} = \sqrt{3x + 4}\).Squaring both sides, we get:
\((2 \sqrt{x})^2 = (\sqrt{3x + 4})^2\).This simplifies to:\[4x = 3x + 4\].

Now our square roots are gone, leaving us with a simpler problem to solve in the next steps.
Isolating Variable
After removing the square roots by squaring both sides, the next step is to isolate the variable. In our simplified equation from the previous step鈥擻[4x = 3x + 4\]鈥攖he goal is to have \(x\) alone on one side of the equation.

Start by subtracting \(3x\) from both sides:\[4x - 3x = 3x + 4 - 3x\].This step gives us:\[x = 4\].

Now, the variable is isolated and we have a potential solution, \(x = 4\). Always remember to check your solutions, especially when dealing with square root equations.
Verifying Solution
The final crucial step is to verify that our solution, \(x = 4\), actually satisfies the original equation. This ensures accuracy, as squaring both sides can sometimes introduce extraneous solutions.

Substitute \(x = 4\) back into the original equation:\[2 \sqrt{4} = \sqrt{3(4) + 4}\].Simplifying further, it becomes:\[2 \cdot 2 = \sqrt{12 + 4}\] or \[4 = \sqrt{16}\].Since \(\sqrt{16} = 4\), both sides of the original equation match, confirming that \(x = 4\) is indeed a correct solution.

By verifying the solution, we ensure that our steps were correct and that the solution fits the original equation without any errors.

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Most popular questions from this chapter

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