Solve For $u$.$4+\frac{5}{u-3}=-\frac{3}{u-2}$If There Is More Than One Solution, Separate Them With Commas. If There Is No Solution, Indicate No Solution.$u=$No Solution

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Introduction

Solving equations involving fractions can be challenging, but with the right approach, we can find the solution. In this problem, we are given the equation $4+\frac{5}{u-3}=-\frac{3}{u-2}$ and we need to solve for $u$. This equation involves fractions, so we will need to use algebraic techniques to simplify and solve it.

Step 1: Multiply both sides of the equation by the least common multiple (LCM) of the denominators

The LCM of the denominators $(u-3)$ and $(u-2)$ is $(u-3)(u-2)$. We will multiply both sides of the equation by this expression to eliminate the fractions.

(u-3)(u-2)\left(4+\frac{5}{u-3}\right) = (u-3)(u-2)\left(-\frac{3}{u-2}\right)

Step 2: Distribute and simplify the equation

We will distribute the $(u-3)(u-2)$ to both sides of the equation and simplify.

(u-3)(u-2)\left(4\right) + (u-3)(u-2)\left(\frac{5}{u-3}\right) = (u-3)(u-2)\left(-\frac{3}{u-2}\right)

Step 3: Simplify the equation further

We can simplify the equation by canceling out the common factors.

4(u-3)(u-2) + 5(u-2) = -3(u-3)

Step 4: Expand and simplify the equation

We will expand and simplify the equation by multiplying out the terms.

4(u^2-5u+6) + 5(u-2) = -3(u-3)

Step 5: Simplify the equation further

We can simplify the equation by combining like terms.

4u^2-20u+24 + 5u-10 = -3u+9

Step 6: Combine like terms

We will combine the like terms on the left-hand side of the equation.

4u^2-15u+14 = -3u+9

Step 7: Move all terms to one side of the equation

We will move all the terms to the left-hand side of the equation.

4u^2-15u+14 + 3u-9 = 0

Step 8: Simplify the equation further

We can simplify the equation by combining like terms.

4u^2-12u+5 = 0

Step 9: Solve the quadratic equation

We will solve the quadratic equation using the quadratic formula.

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

Step 10: Plug in the values into the quadratic formula

We will plug in the values $a=4$, $b=-12$, and $c=5$ into the quadratic formula.

u = \frac{-(-12) \pm \sqrt{(-12)^2-4(4)(5)}}{2(4)}

Step 11: Simplify the expression under the square root

We will simplify the expression under the square root.

u = \frac{12 \pm \sqrt{144-80}}{8}

Step 12: Simplify the expression further

We can simplify the expression further.

u = \frac{12 \pm \sqrt{64}}{8}

Step 13: Simplify the square root

We will simplify the square root.

u = \frac{12 \pm 8}{8}

Step 14: Solve for $u$

We will solve for $u$ by considering both the positive and negative cases.

u = \frac{12 + 8}{8} \text{ or } u = \frac{12 - 8}{8}

Step 15: Simplify the expressions

We will simplify the expressions.

u = \frac{20}{8} \text{ or } u = \frac{4}{8}

Step 16: Simplify the fractions

We will simplify the fractions.

u = \frac{5}{2} \text{ or } u = \frac{1}{2}

Conclusion

We have solved the equation $4+\frac{5}{u-3}=-\frac{3}{u-2}$ for $u$. The solutions are $u=\frac{5}{2}$ and $u=\frac{1}{2}$.

Introduction

Solving equations with fractions can be challenging, but with the right approach, we can find the solution. In this article, we will answer some common questions about solving equations with fractions.

Q: What is the first step in solving an equation with fractions?

A: The first step in solving an equation with fractions is to multiply both sides of the equation by the least common multiple (LCM) of the denominators. This will eliminate the fractions and make it easier to solve the equation.

Q: How do I find the LCM of the denominators?

A: To find the LCM of the denominators, we need to list the multiples of each denominator and find the smallest multiple that is common to both. For example, if the denominators are $(u-3)$ and $(u-2)$, the LCM would be $(u-3)(u-2)$.

Q: What if the equation has multiple fractions?

A: If the equation has multiple fractions, we can simplify the equation by combining the fractions. We can do this by finding the LCM of the denominators and multiplying both sides of the equation by this expression.

Q: How do I simplify the equation after multiplying by the LCM?

A: After multiplying by the LCM, we can simplify the equation by canceling out common factors and combining like terms. This will make it easier to solve the equation.

Q: What if the equation has no solution?

A: If the equation has no solution, it means that the equation is inconsistent and cannot be solved. This can happen if the equation is a contradiction, such as $0=1$.

Q: How do I know if an equation has no solution?

A: We can determine if an equation has no solution by checking if the equation is a contradiction. If the equation is a contradiction, it means that the equation has no solution.

Q: What is the quadratic formula?

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

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

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula, we need to plug in the values of $a$, $b$, and $c$ into the formula. We can then simplify the expression under the square root and solve for $u$.

Q: What if the quadratic formula gives me two solutions?

A: If the quadratic formula gives us two solutions, it means that the equation has two distinct solutions. We can write the solutions as $u=u_1$ and $u=u_2$, where $u_1$ and $u_2$ are the two solutions.

Q: What if the quadratic formula gives me one solution?

A: If the quadratic formula gives us one solution, it means that the equation has one repeated solution. We can write the solution as $u=u_1$, where $u_1$ is the solution.

Q: What if the quadratic formula gives me no solution?

A: If the quadratic formula gives us no solution, it means that the equation has no solution. This can happen if the expression under the square root is negative.

Conclusion

Solving equations with fractions can be challenging, but with the right approach, we can find the solution. By following the steps outlined in this article, we can solve equations with fractions and find the solution.