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Chapter 7: Problem 24

Solve. $$ 3 x^{1 / 2}+12=9 $$

Short Answer

Expert verified
The equation has no real number solution.

Step by step solution

01

- Isolate the square root term

Subtract 12 from both sides of the equation to isolate the term containing the square root.\[ 3x^{1/2} + 12 - 12 = 9 - 12 \]Simplifying both sides:\[ 3x^{1/2} = -3 \]
02

- Solve for the square root term

Divide both sides of the equation by 3 to further isolate the square root term.\[ \frac{3x^{1/2}}{3} = \frac{-3}{3} \]Simplifying both sides:\[ x^{1/2} = -1 \]
03

- Understand the square root concept

Since the square root of a number cannot be negative, there is no real number solution for \( x^{1/2} = -1 \). Hence, the equation has no solution in the real numbers.

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Key Concepts

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

Isolating square root terms
To solve equations involving square roots, it's important to first isolate the square root term. This means getting the term with the square root by itself on one side of the equation.

In our example, we start with the equation: \[ 3 x^{1 / 2} + 12 = 9 \]
We subtract 12 from both sides to isolate the square root term: \[ 3 x^{1 / 2} + 12 - 12 = 9 - 12 \]
Which simplifies to: \[ 3 x^{1 / 2} = -3 \]
This is a crucial step because it simplifies our problem and makes it easier to deal with the square root.
Simplifying Equations
Once the square root term is isolated, the next step is to simplify the equation. This involves further reducing the equation to its simplest form by performing appropriate operations.

In our example, we have: \[ 3 x^{1 / 2} = -3 \]
To simplify, we divide both sides by 3: \[ \frac{3 x^{1 / 2}}{3} = \frac{-3}{3} \]
Which simplifies to: \[ x^{1 / 2} = -1 \]
By simplifying the equation like this, we make it clearer and easier to solve the root term.
No real solution
A square root equation can sometimes have no real solution. This occurs when attempting to find the square root results in a negative number, which is impossible in the set of real numbers.

At our final simplified step, we reached: \[ x^{1 / 2} = -1 \]
The square root of any real number cannot be negative. Therefore: \[ x^{1 / 2} = -1 \] has no real solution.
This is an important concept in understanding square root equations. When solving equations, always check the feasibility of your solution within real numbers to determine if it's valid.

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