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Chapter 9: Problem 3

Solve the equation. $$\sqrt{3 x+4}-2=3$$

Short Answer

Expert verified
The value of x that satisfies the equation is 7

Step by step solution

01

Isolate the square root term

We first isolate the square root term on one side of the equation. \[ \sqrt{3x + 4} = 2 + 3 \] This simplifies to \[ \sqrt{3x + 4} = 5 \]
02

Square both sides of the equation

Now, to eliminate the square root, square both sides of the equation. This gives us \[ (3x + 4) = 5^2 \] Which simplifies further to \[ 3x + 4 = 25 \]
03

Solve for x

Now, the equation can be solved for x by first subtracting 4 from both sides to get \[ 3x = 21 \] Then, divide both sides by 3, this yields \[ x = 7 \]

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Key Concepts

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

Isolate the Square Root
Isolating the square root means getting the square root expression by itself on one side of the equation. It sets us up to solve the equation more easily. In our exercise, the square root term is \(\sqrt{3x + 4}\). Initially, it was part of a larger expression: \(\sqrt{3x + 4} - 2 = 3\).
To isolate \(\sqrt{3x + 4}\), we need to remove the \(-2\) first. We can do this by adding 2 to both sides of the equation:
  • Left Side: \(\sqrt{3x + 4} - 2 + 2 = \sqrt{3x + 4}\)
  • Right Side: \(3 + 2 = 5\)
This results in the simplified equation: \(\sqrt{3x + 4} = 5\).
Remember, isolating the square root is a critical step because it reveals the true value that the square root represents in the equation.
Squaring Both Sides
Once the square root is isolated, the next step is to remove it, which requires squaring both sides of the equation. By squaring, we transform a radical equation into a polynomial one, making it easier to solve.
For the isolated equation \(\sqrt{3x + 4} = 5\), squaring both sides does the following:
  • Left Side: \((\sqrt{3x + 4})^2 = 3x + 4\) because the square of a square root cancels out the radical.
  • Right Side: \(5^2 = 25\)
This results in the equation \(3x + 4 = 25\).
The key takeaway from this step is that squaring both sides effectively removes the square root, simplifying the radical equation into a straightforward algebraic one.
Equation Simplification
Equation simplification is the process of breaking down an equation to its simplest form. This step allows us to find the value of the unknown variable with ease.
From the equation \(3x + 4 = 25\), we perform the following simplifications to solve for \(x\):
  • Subtract 4 from both sides: \(3x + 4 - 4 = 25 - 4\) becomes \(3x = 21\).
  • Divide both sides by 3: \(\frac{3x}{3} = \frac{21}{3}\) simplifies to \(x = 7\).
Through these steps, the equation is fully simplified, and the solution \(x = 7\) is found. Remember that simplifying the equation is crucial for accurate and efficient problem solving.

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Most popular questions from this chapter

Solve the equation. $$\sqrt{15-4 u}-4 \sqrt{u}=0$$

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Rationalize the denominator of the expression and simplify. $$\frac{\sqrt{5}+4}{\sqrt{7}-\sqrt{2}}$$

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