Solving Equations With Square Roots Find The Real Solutions

Finding the real solutions of equations is a fundamental skill in mathematics. This article delves into a step-by-step approach to solve the equation ${ 3\sqrt{3x - 2} - 5 = -4 }$, providing a clear and concise explanation for each step. Whether you're a student tackling algebra problems or a math enthusiast looking to sharpen your skills, this guide will equip you with the knowledge and techniques to solve similar equations effectively.

Understanding the Equation

Before diving into the solution, it's crucial to understand the equation we're dealing with. The given equation is ${ 3\sqrt{3x - 2} - 5 = -4 }$. This equation involves a square root, which means we need to be mindful of the domain of the variable ${ x }$. Specifically, the expression inside the square root, ${ 3x - 2 }$, must be greater than or equal to zero to yield real solutions. This constraint will be important when we check our final answers.

Isolating the Square Root

The first step in solving this equation is to isolate the square root term. This involves performing algebraic manipulations to get the term with the square root on one side of the equation and all other terms on the other side. By isolating the square root, we set the stage for eliminating the radical and simplifying the equation.

To isolate the square root, we start by adding 5 to both sides of the equation:

${ 3\sqrt{3x - 2} - 5 + 5 = -4 + 5 }$

This simplifies to:

${ 3\sqrt{3x - 2} = 1 }$

Next, we divide both sides by 3 to further isolate the square root:

${ \frac{3\sqrt{3x - 2}}{3} = \frac{1}{3} }$

This gives us:

${ \sqrt{3x - 2} = \frac{1}{3} }$

Now that we have isolated the square root, we can proceed to the next step, which is eliminating the square root by squaring both sides of the equation.

Eliminating the Square Root

With the square root isolated, the next step is to eliminate it. This is achieved by squaring both sides of the equation. Squaring both sides ensures that we maintain the equality while removing the radical, allowing us to work with a simpler equation.

Squaring both sides of ${ \sqrt{3x - 2} = \frac{1}{3} }$, we get:

${ (\sqrt{3x - 2})^2 = \left(\frac{1}{3}\right)^2 }$

This simplifies to:

${ 3x - 2 = \frac{1}{9} }$

Now we have a linear equation in terms of ${ x }$, which is much easier to solve. The next step involves solving this linear equation to find the potential value(s) of ${ x }$.

Solving the Linear Equation

After eliminating the square root, we are left with a linear equation: ${ 3x - 2 = \frac{1}{9} }$. Solving this equation involves isolating ${ x }$ by performing algebraic operations. This will give us the potential solution(s) to the original equation.

First, we add 2 to both sides of the equation:

${ 3x - 2 + 2 = \frac{1}{9} + 2 }$

To add ${ \frac{1}{9} }$ and 2, we need to express 2 as a fraction with a denominator of 9. So, 2 becomes ${ \frac{18}{9} }$. The equation now looks like this:

${ 3x = \frac{1}{9} + \frac{18}{9} }$

Adding the fractions, we get:

${ 3x = \frac{19}{9} }$

Next, we divide both sides by 3 to solve for ${ x }$:

${ \frac{3x}{3} = \frac{\frac{19}{9}}{3} }$

Dividing by 3 is the same as multiplying by ${ \frac{1}{3} }$, so we have:

${ x = \frac{19}{9} \cdot \frac{1}{3} }$

This simplifies to:

${ x = \frac{19}{27} }$

So, we have found a potential solution for ${ x }$. However, it is crucial to verify this solution in the original equation to ensure it is a valid solution. This is because squaring both sides of an equation can sometimes introduce extraneous solutions.

Verifying the Solution

After finding a potential solution, it's essential to verify it in the original equation. This step ensures that the solution is valid and not an extraneous solution introduced during the process of squaring both sides. The original equation is ${ 3\sqrt{3x - 2} - 5 = -4 }$, and our potential solution is ${ x = \frac{19}{27} }$.

We substitute ${ x = \frac{19}{27} }$ into the original equation:

${ 3\sqrt{3\left(\frac{19}{27}\right) - 2} - 5 = -4 }$

First, simplify the expression inside the square root:

${ 3\left(\frac{19}{27}\right) - 2 = \frac{19}{9} - 2 }$

Convert 2 to a fraction with a denominator of 9:

${ 2 = \frac{18}{9} }$

So, the expression becomes:

${ \frac{19}{9} - \frac{18}{9} = \frac{1}{9} }$

Now, substitute this back into the equation:

${ 3\sqrt{\frac{1}{9}} - 5 = -4 }$

The square root of ${ \frac{1}{9} }$ is ${ \frac{1}{3} }$, so we have:

${ 3\left(\frac{1}{3}\right) - 5 = -4 }$

Simplify:

${ 1 - 5 = -4 }$

${ -4 = -4 }$

Since the equation holds true, ${ x = \frac{19}{27} }$ is a valid solution.

Considering the Domain

Before finalizing our solution, we must consider the domain of the original equation. The expression inside the square root, ${ 3x - 2 }$, must be greater than or equal to zero for the solution to be real. This gives us the inequality:

${ 3x - 2 \geq 0 }$

Add 2 to both sides:

${ 3x \geq 2 }$

Divide by 3:

${ x \geq \frac{2}{3} }$

Now, we need to check if our solution, ${ x = \frac{19}{27} }$, satisfies this condition. Convert ${ \frac{2}{3} }$ to a fraction with a denominator of 27:

${ \frac{2}{3} = \frac{2 \cdot 9}{3 \cdot 9} = \frac{18}{27} }$

So, the condition is ${ x \geq \frac{18}{27} }$. Since ${ \frac{19}{27} }$ is greater than ${ \frac{18}{27} }$, our solution satisfies the domain condition.

Conclusion

In conclusion, the real solution to the equation ${ 3\sqrt{3x - 2} - 5 = -4 }$ is ${ x = \frac{19}{27} }$. We arrived at this solution by systematically isolating the square root, eliminating it by squaring both sides, solving the resulting linear equation, and verifying the solution in the original equation. Additionally, we considered the domain of the equation to ensure the solution was valid. This comprehensive approach ensures accuracy and a thorough understanding of the problem-solving process. By mastering these steps, you can confidently tackle similar equations and enhance your mathematical skills.

This article provided a detailed, step-by-step solution to finding the real solutions of the equation ${ 3\sqrt{3x - 2} - 5 = -4 }$. By understanding each step, from isolating the square root to verifying the solution, readers can apply these techniques to solve a wide range of similar equations. Remember, the key to mastering mathematics is practice and a thorough understanding of the underlying principles.