How Many Real Solutions Does The Equation 3 X − 1 = 2 X − 5 \sqrt{3x-1} = 2x-5 3 X − 1 ​ = 2 X − 5 Have?A. 0 B. 1 C. 2 D. Cannot Be Determined From The Graph

Introduction

When it comes to solving equations, one of the most crucial aspects is determining the number of real solutions. In this article, we will delve into the equation $\sqrt{3x-1} = 2x-5$ and explore the process of finding the number of real solutions. We will examine the equation, identify the steps to solve it, and finally, determine the number of real solutions.

Understanding the Equation

The given equation is $\sqrt{3x-1} = 2x-5$. To begin solving this equation, we need to understand the properties of square roots. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we have $\sqrt{3x-1}$, which means that the value inside the square root, $3x-1$, must be non-negative.

Solving the Equation

To solve the equation, we can start by isolating the square root term. We can do this by squaring both sides of the equation. Squaring both sides will eliminate the square root, allowing us to solve for $x$.

(\sqrt{3x-1})^2 = (2x-5)^2

Expanding both sides of the equation, we get:

3x-1 = 4x^2 - 20x + 25

Rearranging the terms, we get a quadratic equation:

4x^2 - 23x + 26 = 0

Solving the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 4$, $b = -23$, and $c = 26$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-23) \pm \sqrt{(-23)^2 - 4(4)(26)}}{2(4)}$$

Simplifying the expression, we get:

$$x = \frac{23 \pm \sqrt{529 - 416}}{8}$$

$$x = \frac{23 \pm \sqrt{113}}{8}$$

Analyzing the Solutions

Now that we have the solutions to the quadratic equation, we need to analyze them to determine the number of real solutions. The solutions are given by:

$$x = \frac{23 \pm \sqrt{113}}{8}$$

Since $\sqrt{113}$ is a positive value, both solutions will be real numbers. However, we need to check if the solutions satisfy the original equation.

Checking the Solutions

To check the solutions, we need to plug them back into the original equation. Let's start with the first solution:

$$x = \frac{23 + \sqrt{113}}{8}$$

Plugging this value into the original equation, we get:

$$\sqrt{3x-1} = 2x-5$$

$$\sqrt{3\left(\frac{23 + \sqrt{113}}{8}\right)-1} = 2\left(\frac{23 + \sqrt{113}}{8}\right)-5$$

Simplifying the expression, we get:

$$\sqrt{\frac{69 + 3\sqrt{113}}{8}-1} = \frac{23 + \sqrt{113}}{4}-5$$

$$\sqrt{\frac{61 + 3\sqrt{113}}{8}} = \frac{23 + \sqrt{113}}{4}-5$$

Since the left-hand side is a square root, it must be non-negative. However, the right-hand side is a negative value. This means that the first solution does not satisfy the original equation.

Conclusion

In conclusion, we have analyzed the equation $\sqrt{3x-1} = 2x-5$ and determined that it has no real solutions. The solutions to the quadratic equation are real numbers, but they do not satisfy the original equation. Therefore, the correct answer is A. 0.

Final Thoughts

Solving equations can be a complex process, and it requires careful analysis and attention to detail. In this article, we have demonstrated the importance of checking the solutions to ensure that they satisfy the original equation. By following these steps, we can determine the number of real solutions to an equation and gain a deeper understanding of the underlying mathematics.

Introduction

When it comes to solving equations, one of the most crucial aspects is determining the number of real solutions. In this article, we will delve into the equation $\sqrt{3x-1} = 2x-5$ and explore the process of finding the number of real solutions. We will examine the equation, identify the steps to solve it, and finally, determine the number of real solutions.

Understanding the Equation

The given equation is $\sqrt{3x-1} = 2x-5$. To begin solving this equation, we need to understand the properties of square roots. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we have $\sqrt{3x-1}$, which means that the value inside the square root, $3x-1$, must be non-negative.

Solving the Equation

To solve the equation, we can start by isolating the square root term. We can do this by squaring both sides of the equation. Squaring both sides will eliminate the square root, allowing us to solve for $x$.

(\sqrt{3x-1})^2 = (2x-5)^2

Expanding both sides of the equation, we get:

3x-1 = 4x^2 - 20x + 25

Rearranging the terms, we get a quadratic equation:

4x^2 - 23x + 26 = 0

Solving the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 4$, $b = -23$, and $c = 26$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-23) \pm \sqrt{(-23)^2 - 4(4)(26)}}{2(4)}$$

Simplifying the expression, we get:

$$x = \frac{23 \pm \sqrt{529 - 416}}{8}$$

$$x = \frac{23 \pm \sqrt{113}}{8}$$

Analyzing the Solutions

Now that we have the solutions to the quadratic equation, we need to analyze them to determine the number of real solutions. The solutions are given by:

$$x = \frac{23 \pm \sqrt{113}}{8}$$

Since $\sqrt{113}$ is a positive value, both solutions will be real numbers. However, we need to check if the solutions satisfy the original equation.

Checking the Solutions

To check the solutions, we need to plug them back into the original equation. Let's start with the first solution:

$$x = \frac{23 + \sqrt{113}}{8}$$

Plugging this value into the original equation, we get:

$$\sqrt{3x-1} = 2x-5$$

$$\sqrt{3\left(\frac{23 + \sqrt{113}}{8}\right)-1} = 2\left(\frac{23 + \sqrt{113}}{8}\right)-5$$

Simplifying the expression, we get:

$$\sqrt{\frac{69 + 3\sqrt{113}}{8}-1} = \frac{23 + \sqrt{113}}{4}-5$$

$$\sqrt{\frac{61 + 3\sqrt{113}}{8}} = \frac{23 + \sqrt{113}}{4}-5$$

Since the left-hand side is a square root, it must be non-negative. However, the right-hand side is a negative value. This means that the first solution does not satisfy the original equation.

Conclusion

In conclusion, we have analyzed the equation $\sqrt{3x-1} = 2x-5$ and determined that it has no real solutions. The solutions to the quadratic equation are real numbers, but they do not satisfy the original equation. Therefore, the correct answer is A. 0.

Q&A

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to isolate the square root term by squaring both sides of the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula used to solve quadratic equations. It is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: What is the significance of the square root in the equation?

A: The square root in the equation represents a value that, when multiplied by itself, gives the original number. In this case, the value inside the square root, $3x-1$, must be non-negative.

Q: How do we check the solutions to the equation?

A: To check the solutions, we need to plug them back into the original equation and simplify the expression. If the left-hand side is a square root, it must be non-negative. If the right-hand side is a negative value, the solution does not satisfy the original equation.

Q: What is the final answer to the equation?

A: The final answer to the equation is A. 0, which means that the equation has no real solutions.

Q: What is the importance of analyzing the solutions to the equation?

A: Analyzing the solutions to the equation is crucial in determining the number of real solutions. By checking the solutions, we can ensure that they satisfy the original equation and provide a valid solution.

Q: What is the significance of the quadratic equation in solving the equation?

A: The quadratic equation is a crucial step in solving the equation. By using the quadratic formula, we can find the solutions to the equation and analyze them to determine the number of real solutions.

Q: What is the final thought on solving the equation?

A: Solving equations can be a complex process, and it requires careful analysis and attention to detail. By following these steps, we can determine the number of real solutions to an equation and gain a deeper understanding of the underlying mathematics.