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Chapter 0: Problem 115

Solve each radical equation. Check all proposed solutions. $$\sqrt{3 x+18}=x$$

Short Answer

Expert verified
For the given equation, the valid solution is \( x=6 \).

Step by step solution

01

Isolate Radical Expression

The first step is to get the radical all by itself on one side of the equation. In this case, this is already done: \( \sqrt{3x + 18} = x \). This equation tells that square root of \(3x + 18\) results in \(x\).
02

Square Both Sides

Squaring both sides of the equation to eliminate the square root. We get \( (\sqrt{3x + 18})^2 = x^2 \). Or, it simplifies to \( 3x + 18 = x^2 \).
03

Convert to Quadratic Equation

Since we have \(x^2\) in the equation, it's best to convert it into a quadratic equation to find possible values of \(x\). This is done by bringing the entire equation to one side. Thus the equation is now \( x^2 - 3x - 18 = 0 \).
04

Factor Quadratic Equation

The given equation \( x^2 - 3x - 18 = 0 \) factors to \( (x-6)(x+3)=0 \). The factoring is done as -6 and 3 are the numbers which added gives -3 and multiplied gives -18.
05

Find Possible Solutions

Once the equation is factored, we can set each of the factors equal to zero and solve for \(x\). Hence \( x-6=0 \) giving \( x=6 \), and \( x+3=0 \) giving \( x=-3 \). Both 6 and -3 are the proposed solutions.
06

Validate Solutions

Finally, substitute the values into the original equation to ensure they satisfy the equation. For \( x=6 \), substituting \( x \) in \( \sqrt{3x + 18} = x \) gives \( \sqrt{3(6) + 18} = 6 \), which simplifies to \( 6=6 \), a true statement. For \( x=-3 \), substituting \( x \) gives \( \sqrt{3(-3) + 18} = -3 \), which simplifies to \( \sqrt{9} = -3 \) or \( 3=-3 \), a false statement thus \( x=-3 \) is an extraneous solution and is discarded. So, the final solution is \( x=6 \).

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Most popular questions from this chapter

Exercises \(142-144\) will help you prepare for the material covered in the next section. Use the distributive property to multiply: $$ 2 x^{4}\left(8 x^{4}+3 x\right) $$

Describe how to solve an absolute value inequality involving the symbol \(>.\) Give an example.

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x,\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -3 and ending with 3

Perform the indicated operations. Simplify the result, if possible. $$\frac{y^{-1}-(y+2)^{-1}}{2}$$

This will help you prepare for the material covered in the next section. Multiply and simplify: \(12\left(\frac{x+2}{4}-\frac{x-1}{3}\right)\).
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