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Chapter 2: Problem 81

\(3 \sqrt{x}-6=0\)

Short Answer

Expert verified
From the above step by step solution, the value of \(x\) is \(4\).

Step by step solution

01

Move the Constant to the other Side

The first step is to isolate the square root term. To do so, add 6 to both sides of the equation. You get \(3 \sqrt{x}=6\)
02

Isolate the square root

Now you need to get rid \(3\) which is multiplying the square root. Divide both sides of the equation by 3. You get \(\sqrt{x}=2\)
03

Square both sides

To get rid of the square root on one side, square both sides of the equation. This will give you \(x=4\)

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Key Concepts

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

Isolate the Square Root
When solving equations involving square roots, a crucial initial step is to isolate the square root term. This provides clarity and prepares the equation for subsequent steps. Here's how it's done:

  • Identify the square root term in the equation.
  • Perform inverse operations to move all other terms to the opposite side of the equation.
  • The aim is to have the square root term alone on one side of the equation.

In our example, moving the constant involves adding 6 to both sides of the equation, simplifying it to \(3 \sqrt{x}=6\). Next, to fully isolate \(\sqrt{x}\), we divide both sides by 3, resulting in \(\sqrt{x}=2\). By isolating the square root, we've set the stage for easier further manipulation.
Square Both Sides of an Equation
Once the square root is isolated, the next logical step is to eliminate the root. This is accomplished by squaring both sides of the equation. Squaring is the reverse operation to taking a square root, thus it neutralizes the square root and simplifies the expression.

Here's what you need to know about squaring both sides:
  • Ensure that the square root is alone before squaring to avoid complicating the equation with added squares.
  • Whatever you do to one side of an equation, do to the other to maintain balance.

Applying this to our example, when we square \(\sqrt{x}\) and 2, we get \(x=2^2\) or \(x=4\). It's critical to remember that while squaring gets rid of the root, it may introduce additional solutions, known as extraneous solutions, which we have to check for validity at the end.
Solving Radical Equations
Solving radical equations, which include square roots, involves a series of strategic steps. The overarching goal is to isolate the radical expression and then remove it, revealing the solution.

Here's a simplified process:
  • Isolate the radical expression by moving all other terms to the other side.
  • Square both sides of the equation to eliminate the square root.
  • Solve the resulting equation just as you would a regular linear or quadratic equation.
  • Check your solutions by substituting them back into the original equation to ensure they don't produce any contradictions.

In practice, our example illustrates this process effectively. We isolate \(\sqrt{x}\), square both sides to find \(x=4\), and finally, substitute this solution into the original equation to verify its correctness: \(3 \sqrt{4}-6=0\), which simplifies to \(3*2-6=0\) and verifies that \(x=4\) is indeed a solution.

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Most popular questions from this chapter

MODELING WITH MATHEMATICS The Gateshead Millennium Bridge spans the River Tyne. The arch of the bridge can be modeled by a parabola. The arch reaches a maximum height of 50 meters at a point roughly 63 meters across the river. Graph the curve of the arch. What are the domain and range? What do they represent in this situation?

\(h(x)=2 x^2-12 x\)

\(f(x)=(x-3)^2\)

\(g(x)=2(x-1)^2-2\)

\(f(x)=-2(x-1)^2-5\)
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