Solve For { X $}$ In The Equation:${ \sqrt{3x^2 - 4x + 34} + \sqrt{3x^2 - 4x - 11} = 9 }$

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Introduction

In this article, we will delve into solving a quadratic equation that involves square roots. The given equation is $\sqrt{3x^2 - 4x + 34} + \sqrt{3x^2 - 4x - 11} = 9$. Our goal is to find the value of $x$ that satisfies this equation. We will break down the solution step by step, using algebraic manipulations and properties of square roots.

Step 1: Understand the Equation

The given equation involves two square roots, each of which is a quadratic expression. To simplify the equation, we can start by isolating one of the square roots. Let's isolate the first square root on the left-hand side of the equation.

$\sqrt{3x^2 - 4x + 34} + \sqrt{3x^2 - 4x - 11} = 9$

$\sqrt{3x^2 - 4x + 34} = 9 - \sqrt{3x^2 - 4x - 11}$

Step 2: Square Both Sides of the Equation

To eliminate the square root, we can square both sides of the equation. This will allow us to simplify the equation and make it easier to solve.

$(\sqrt{3x^2 - 4x + 34})^2 = (9 - \sqrt{3x^2 - 4x - 11})^2$

$3x^2 - 4x + 34 = 81 - 18\sqrt{3x^2 - 4x - 11} + (3x^2 - 4x - 11)$

Step 3: Simplify the Equation

Now that we have squared both sides of the equation, we can simplify it by combining like terms.

$3x^2 - 4x + 34 = 81 - 18\sqrt{3x^2 - 4x - 11} + 3x^2 - 4x - 11$

$3x^2 - 4x + 34 - 3x^2 + 4x + 11 = 81 - 18\sqrt{3x^2 - 4x - 11}$

$45 = 81 - 18\sqrt{3x^2 - 4x - 11}$

Step 4: Isolate the Square Root

Now that we have simplified the equation, we can isolate the square root on one side of the equation.

$18\sqrt{3x^2 - 4x - 11} = 81 - 45$

$18\sqrt{3x^2 - 4x - 11} = 36$

$\sqrt{3x^2 - 4x - 11} = 2$

Step 5: Square Both Sides of the Equation Again

To eliminate the square root, we can square both sides of the equation again.

$(\sqrt{3x^2 - 4x - 11})^2 = 2^2$

$3x^2 - 4x - 11 = 4$

Step 6: Solve for $x$

Now that we have simplified the equation, we can solve for $x$.

$3x^2 - 4x - 11 = 4$

$3x^2 - 4x - 15 = 0$

Step 7: Factor the Quadratic Equation

To solve for $x$, we can factor the quadratic equation.

$(3x + 5)(x - 3) = 0$

Step 8: Solve for $x$

Now that we have factored the quadratic equation, we can solve for $x$.

$3x + 5 = 0$

$x = -\frac{5}{3}$

$x - 3 = 0$

$x = 3$

Conclusion

In this article, we have solved the equation $\sqrt{3x^2 - 4x + 34} + \sqrt{3x^2 - 4x - 11} = 9$ using algebraic manipulations and properties of square roots. We have broken down the solution into several steps, including isolating the square root, squaring both sides of the equation, simplifying the equation, and solving for $x$. The final solution is $x = -\frac{5}{3}$ or $x = 3$.

Final Answer

The final answer is $\boxed{-\frac{5}{3}, 3}$.

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Introduction

In our previous article, we solved the equation $\sqrt{3x^2 - 4x + 34} + \sqrt{3x^2 - 4x - 11} = 9$ using algebraic manipulations and properties of square roots. In this article, we will answer some frequently asked questions about the solution.

Q: What is the main concept behind solving this equation?

A: The main concept behind solving this equation is to isolate the square root, square both sides of the equation, simplify the equation, and solve for $x$. This involves using algebraic manipulations and properties of square roots.

Q: Why do we need to isolate the square root?

A: We need to isolate the square root to eliminate it from the equation. By isolating the square root, we can square both sides of the equation and simplify it.

Q: What is the significance of squaring both sides of the equation?

A: Squaring both sides of the equation allows us to eliminate the square root and simplify the equation. This is a crucial step in solving the equation.

Q: How do we simplify the equation after squaring both sides?

A: After squaring both sides of the equation, we can simplify it by combining like terms. This involves rearranging the terms and eliminating any unnecessary terms.

Q: What is the final solution to the equation?

A: The final solution to the equation is $x = -\frac{5}{3}$ or $x = 3$.

Q: Can you explain the steps involved in solving the equation?

A: Here are the steps involved in solving the equation:

1. Isolate the square root
2. Square both sides of the equation
3. Simplify the equation
4. Solve for $x$

Q: What are some common mistakes to avoid when solving this equation?

A: Some common mistakes to avoid when solving this equation include:

• Not isolating the square root
• Not squaring both sides of the equation
• Not simplifying the equation
• Not solving for $x$

Q: Can you provide some examples of similar equations that can be solved using the same method?

A: Yes, here are some examples of similar equations that can be solved using the same method:

• $\sqrt{2x^2 - 5x + 12} + \sqrt{2x^2 - 5x - 3} = 7$
• $\sqrt{x^2 - 4x + 5} + \sqrt{x^2 - 4x - 2} = 3$
• $\sqrt{3x^2 - 2x + 1} + \sqrt{3x^2 - 2x - 4} = 5$

Conclusion

In this article, we have answered some frequently asked questions about solving the equation $\sqrt{3x^2 - 4x + 34} + \sqrt{3x^2 - 4x - 11} = 9$. We have provided explanations and examples to help clarify the concepts involved in solving the equation.

Final Answer

The final answer is $\boxed{-\frac{5}{3}, 3}$.