Solve For $u$.$\frac{2u}{u-2} = -\frac{8}{u^2 - 6u + 8}$If There Is More Than One Solution, Separate Them With Commas. If There Is No Solution, Click On No Solution.$u = \square$\text{No Solution}$

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Introduction

Solving equations involving fractions can be a challenging task, especially when the fractions are complex. In this case, we are given an equation with a fraction on both sides, and we need to solve for the variable $u$. The equation is $\frac{2u}{u-2} = -\frac{8}{u^2 - 6u + 8}$. Our goal is to isolate the variable $u$ and find its value.

Step 1: Simplify the Right-Hand Side of the Equation

To simplify the right-hand side of the equation, we need to factor the denominator $u^2 - 6u + 8$. We can factor it as $(u-2)(u-4)$. Therefore, the equation becomes $\frac{2u}{u-2} = -\frac{8}{(u-2)(u-4)}$.

Step 2: Eliminate the Fraction on the Left-Hand Side

To eliminate the fraction on the left-hand side, we can multiply both sides of the equation by the denominator of the left-hand side, which is $u-2$. This gives us $2u = -\frac{8}{u-4}$.

Step 3: Eliminate the Fraction on the Right-Hand Side

To eliminate the fraction on the right-hand side, we can multiply both sides of the equation by the denominator of the right-hand side, which is $u-4$. This gives us $2u(u-4) = -8$.

Step 4: Expand and Simplify the Equation

Expanding the left-hand side of the equation, we get $2u^2 - 8u = -8$. Simplifying the equation, we get $2u^2 - 8u + 8 = 0$.

Step 5: Solve the Quadratic Equation

To solve the quadratic equation $2u^2 - 8u + 8 = 0$, we can use the quadratic formula: $u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 2$, $b = -8$, and $c = 8$. Plugging these values into the quadratic formula, we get $u = \frac{8 \pm \sqrt{(-8)^2 - 4(2)(8)}}{2(2)}$.

Step 6: Simplify the Quadratic Formula

Simplifying the quadratic formula, we get $u = \frac{8 \pm \sqrt{64 - 64}}{4}$. This simplifies to $u = \frac{8 \pm \sqrt{0}}{4}$.

Step 7: Solve for $u$

Since the square root of 0 is 0, we have $u = \frac{8 \pm 0}{4}$. This simplifies to $u = \frac{8}{4}$.

Step 8: Simplify the Solution

Simplifying the solution, we get $u = 2$.

Conclusion

In this problem, we were given an equation involving a fraction on both sides. We simplified the right-hand side of the equation, eliminated the fractions on both sides, expanded and simplified the equation, and solved the quadratic equation using the quadratic formula. The solution to the equation is $u = 2$.

Discussion

The solution to the equation $u = 2$ is a single value. However, in some cases, an equation may have multiple solutions or no solution at all. In this case, we found a single solution, which is $u = 2$. If we had found multiple solutions, we would have separated them with commas. If we had found no solution, we would have clicked on "No solution".

Final Answer

The final answer is $\boxed{2}$.

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Introduction

In our previous article, we solved the equation $\frac{2u}{u-2} = -\frac{8}{u^2 - 6u + 8}$ and found that the solution is $u = 2$. However, we may have some questions about the solution and the steps involved in solving the equation. In this article, we will answer some of the frequently asked questions about the solution and the steps involved in solving the equation.

Q: What is the solution to the equation $\frac{2u}{u-2} = -\frac{8}{u^2 - 6u + 8}$?

A: The solution to the equation is $u = 2$.

Q: Why did we simplify the right-hand side of the equation?

A: We simplified the right-hand side of the equation to make it easier to eliminate the fractions on both sides of the equation.

Q: Why did we multiply both sides of the equation by the denominator of the left-hand side?

A: We multiplied both sides of the equation by the denominator of the left-hand side to eliminate the fraction on the left-hand side.

Q: Why did we multiply both sides of the equation by the denominator of the right-hand side?

A: We multiplied both sides of the equation by the denominator of the right-hand side to eliminate the fraction on the right-hand side.

Q: Why did we expand and simplify the equation?

A: We expanded and simplified the equation to make it easier to solve the quadratic equation.

Q: How did we solve the quadratic equation?

A: We solved the quadratic equation using the quadratic formula: $u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations of the form $ax^2 + bx + c = 0$. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

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.

Q: What is the final answer to the equation?

A: The final answer to the equation is $u = 2$.

Q: Is the solution to the equation unique?

A: Yes, the solution to the equation is unique.

Q: Can the equation have multiple solutions?

A: No, the equation cannot have multiple solutions.

Q: Can the equation have no solution?

A: No, the equation cannot have no solution.

Conclusion

In this article, we answered some of the frequently asked questions about the solution and the steps involved in solving the equation $\frac{2u}{u-2} = -\frac{8}{u^2 - 6u + 8}$. We hope that this article has been helpful in clarifying any doubts that you may have had about the solution and the steps involved in solving the equation.

Final Answer

The final answer is $\boxed{2}$.