/*! 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 13 \(\sqrt{3 x+1}-4=0\)... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 13

\(\sqrt{3 x+1}-4=0\)

Short Answer

Expert verified
x = 5

Step by step solution

01

Isolate the Square Root

Start by isolating the square root expression on one side of the equation. To do this, add 4 to both sides of the equation: \[ \sqrt{3x + 1} - 4 + 4 = 0 + 4 \] \[ \sqrt{3x + 1} = 4 \]
02

Square Both Sides

Next, eliminate the square root by squaring both sides of the equation: \[ (\sqrt{3x + 1})^2 = 4^2 \] \[ 3x + 1 = 16 \]
03

Solve for x

Now, solve for the variable x. Start by subtracting 1 from both sides of the equation to isolate the term with x: \[ 3x + 1 - 1 = 16 - 1 \] \[ 3x = 15 \]Then, divide both sides by 3: \[ x = \frac{15}{3} \] \[ x = 5 \]

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Key Concepts

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

Isolate the Square Root
In the first step of solving square root equations, the goal is to isolate the square root expression. This means getting the square root term alone on one side of the equation.

For example, consider the equation: \(\because \sqrt{3x+1} - 4 = 0\).
To isolate \(\because \sqrt{3x+1} \) , we need to move the constant term on the left side to the right side. We can do this by adding 4 to both sides:

  • Add 4 to both sides: \(\because \sqrt{3x+1} - 4 + 4 = 0 + 4\)
  • This simplifies to: \(\because \sqrt{3x+1} = 4\)
After this step, the square root is isolated on one side, making it easier to work with in the next steps.
Square Both Sides
The next step is to eliminate the square root by squaring both sides of the equation. Squaring the equation helps us to remove the square root sign. This step is crucial because it allows us to transform the equation into a form that can be solved using algebraic methods.

Let's continue with our example where we have: \(\because \sqrt{3x+1} = 4\).

To square both sides:
  • Square the left side: \(\because (\sqrt{3x+1})^2 = 3x + 1\)
  • Square the right side: \(\because 4^2 = 16\)
As a result, we get:
\(\because 3x + 1 = 16\).
This eliminates the square root and transforms the equation into a simpler linear form that can be more easily solved.
Solve for x
In the final step, we solve for the variable \(\because x\). The equation we have after squaring both sides is a standard linear equation: \(\because 3x + 1 = 16\).

To solve for \(\because x\), we perform the following steps:

  • Subtract 1 from both sides: \(\because 3x + 1 - 1 = 16 - 1\)
  • This simplifies to: \(\because 3x = 15\)
  • Divide both sides by 3: \(\because 3x / 3 = 15 / 3\)
  • As a result, we find: \(\because x = 5\)
By following these steps, we isolate \(\because x\) and determine its value. With \(\because x = 5\), we have successfully solved the problem.

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Most popular questions from this chapter

Write each quotient in lowest terms. Assume that all variables represent positive real numbers. $$ \frac{3-3 \sqrt{5}}{3} $$

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