Solve The Equation For \[$ X \$\]:$\[ 4 \sqrt{x-3} - \sqrt{6x-17} = 3 \\]

Introduction

In this article, we will delve into the world of algebra and solve a complex equation involving square roots. The given equation is $4 \sqrt{x-3} - \sqrt{6x-17} = 3$. Our goal is to isolate the variable $x$ and find its value. We will use various algebraic techniques, including isolating the square root terms and squaring both sides of the equation.

Step 1: Isolate the Square Root Terms

The first step in solving this equation is to isolate the square root terms on one side of the equation. We can do this by adding $\sqrt{6x-17}$ to both sides of the equation:

$$4 \sqrt{x-3} = 3 + \sqrt{6x-17}$$

Step 2: Square Both Sides of the Equation

Next, we will square both sides of the equation to eliminate the square root terms. This will give us:

$$(4 \sqrt{x-3})^2 = (3 + \sqrt{6x-17})^2$$

Expanding the right-hand side of the equation, we get:

$$16(x-3) = 9 + 6\sqrt{6x-17} + (6x-17)$$

Step 3: Simplify the Equation

Now, we will simplify the equation by combining like terms:

$$16x - 48 = 9 + 6\sqrt{6x-17} + 6x - 17$$

Subtracting $6x$ from both sides of the equation, we get:

$$10x - 48 = 9 + 6\sqrt{6x-17} - 17$$

Simplifying further, we get:

$$10x - 48 = -8 + 6\sqrt{6x-17}$$

Step 4: Isolate the Square Root Term

Next, we will isolate the square root term by adding $8$ to both sides of the equation:

$$10x - 56 = 6\sqrt{6x-17}$$

Step 5: Square Both Sides of the Equation Again

We will square both sides of the equation again to eliminate the square root term:

$$(10x - 56)^2 = (6\sqrt{6x-17})^2$$

Expanding the left-hand side of the equation, we get:

$$100x^2 - 1120x + 3136 = 36(6x-17)$$

Simplifying the right-hand side of the equation, we get:

$$100x^2 - 1120x + 3136 = 216x - 612$$

Step 6: Simplify the Equation Again

Now, we will simplify the equation by combining like terms:

$$100x^2 - 1120x + 3136 = 216x - 612$$

Subtracting $216x$ from both sides of the equation, we get:

$$100x^2 - 1336x + 3136 = -612$$

Adding $612$ to both sides of the equation, we get:

$$100x^2 - 1336x + 3748 = 0$$

Step 7: Solve the Quadratic Equation

We will solve the quadratic equation using the quadratic formula:

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

In this case, $a = 100$, $b = -1336$, and $c = 3748$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-1336) \pm \sqrt{(-1336)^2 - 4(100)(3748)}}{2(100)}$$

Simplifying the expression under the square root, we get:

$$x = \frac{1336 \pm \sqrt{1782912 - 14992000}}{200}$$

Simplifying further, we get:

$$x = \frac{1336 \pm \sqrt{-13208088}}{200}$$

Since the expression under the square root is negative, we know that there are no real solutions to the equation.

Conclusion

In this article, we solved the equation $4 \sqrt{x-3} - \sqrt{6x-17} = 3$ using various algebraic techniques. We isolated the square root terms, squared both sides of the equation, and simplified the resulting equation. However, we found that the equation has no real solutions. This is because the expression under the square root is negative, which means that there are no real values of $x$ that satisfy the equation.

Final Answer

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Introduction

In our previous article, we solved the equation $4 \sqrt{x-3} - \sqrt{6x-17} = 3$ using various algebraic techniques. However, we found that the equation has no real solutions. In this article, we will answer some frequently asked questions about the equation and provide additional insights into the solution process.

Q: What is the main challenge in solving this equation?

A: The main challenge in solving this equation is the presence of square root terms. The equation involves two square root terms, which makes it difficult to isolate the variable $x$.

Q: Why did we square both sides of the equation twice?

A: We squared both sides of the equation twice to eliminate the square root terms. Squaring both sides of the equation once would have eliminated one of the square root terms, but it would have introduced a new square root term. By squaring both sides of the equation twice, we were able to eliminate both square root terms and simplify the equation.

Q: Why did we use the quadratic formula to solve the equation?

A: We used the quadratic formula to solve the equation because it is a quadratic equation in the form $ax^2 + bx + c = 0$. The quadratic formula is a powerful tool for solving quadratic equations, and it allows us to find the solutions to the equation.

Q: What is the significance of the negative expression under the square root?

A: The negative expression under the square root is significant because it indicates that there are no real solutions to the equation. When the expression under the square root is negative, it means that there are no real values of $x$ that satisfy the equation.

Q: Can we find complex solutions to the equation?

A: Yes, we can find complex solutions to the equation. The quadratic formula can be used to find complex solutions to the equation, and it can be expressed in the form $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. However, in this case, we found that the expression under the square root is negative, which means that there are no real solutions to the equation.

Q: What are some common mistakes to avoid when solving equations involving square roots?

A: Some common mistakes to avoid when solving equations involving square roots include:

• Not isolating the square root terms
• Not squaring both sides of the equation correctly
• Not simplifying the equation correctly
• Not using the quadratic formula correctly

Q: How can we apply the concepts learned in this article to other problems?

A: The concepts learned in this article can be applied to other problems involving square roots and quadratic equations. By understanding how to isolate square root terms, square both sides of the equation, and simplify the resulting equation, we can solve a wide range of problems involving square roots and quadratic equations.

Conclusion

In this article, we answered some frequently asked questions about the equation $4 \sqrt{x-3} - \sqrt{6x-17} = 3$ and provided additional insights into the solution process. We discussed the main challenges in solving the equation, the significance of the negative expression under the square root, and how to apply the concepts learned in this article to other problems. By understanding how to solve equations involving square roots and quadratic equations, we can develop a deeper understanding of algebra and mathematics.

Final Answer

The final answer is: $\boxed{No real solutions}$