Solve The Equation:${ \frac{3}{x-2} + \frac{2}{x+1} = \frac{-5}{1} }$

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Introduction

In this article, we will delve into the world of algebra and focus on solving a complex equation involving fractions. The equation we will be working with is:

3x−2+2x+1=−51{ \frac{3}{x-2} + \frac{2}{x+1} = \frac{-5}{1} }

This equation may seem daunting at first, but with a clear understanding of the steps involved, we can break it down and solve for the variable x.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at what we're dealing with. We have three fractions on the left-hand side of the equation, and a single fraction on the right-hand side. The left-hand side consists of two fractions with different denominators, while the right-hand side has a single fraction with a denominator of 1.

To solve this equation, we need to find a common denominator for the fractions on the left-hand side, and then combine them into a single fraction. Once we have a single fraction on the left-hand side, we can equate it to the fraction on the right-hand side and solve for x.

Finding a Common Denominator

To find a common denominator for the fractions on the left-hand side, we need to identify the least common multiple (LCM) of the denominators. In this case, the denominators are x-2 and x+1. The LCM of these two expressions is (x-2)(x+1).

Now that we have the common denominator, we can rewrite the fractions on the left-hand side with the common denominator:

3(x+1)(x−2)(x+1)+2(x−2)(x−2)(x+1)=−51{ \frac{3(x+1)}{(x-2)(x+1)} + \frac{2(x-2)}{(x-2)(x+1)} = \frac{-5}{1} }

Combining the Fractions

Now that we have the fractions with the common denominator, we can combine them into a single fraction:

3(x+1)+2(x−2)(x−2)(x+1)=−51{ \frac{3(x+1) + 2(x-2)}{(x-2)(x+1)} = \frac{-5}{1} }

To combine the fractions, we need to add the numerators (the top parts of the fractions) and keep the common denominator:

3x+3+2x−4(x−2)(x+1)=−51{ \frac{3x + 3 + 2x - 4}{(x-2)(x+1)} = \frac{-5}{1} }

Simplifying the numerator, we get:

5x−1(x−2)(x+1)=−51{ \frac{5x - 1}{(x-2)(x+1)} = \frac{-5}{1} }

Equating the Fractions

Now that we have a single fraction on the left-hand side, we can equate it to the fraction on the right-hand side:

5x−1(x−2)(x+1)=−51{ \frac{5x - 1}{(x-2)(x+1)} = \frac{-5}{1} }

To equate the fractions, we need to multiply both sides of the equation by the denominator of the right-hand side, which is 1:

5x−1=−5(x−2)(x+1){ 5x - 1 = -5(x-2)(x+1) }

Solving for x

Now that we have the equation in a simpler form, we can solve for x. To do this, we need to expand the right-hand side of the equation and then isolate x:

5x−1=−5(x2+x−2){ 5x - 1 = -5(x^2 + x - 2) }

Expanding the right-hand side, we get:

5x−1=−5x2−5x+10{ 5x - 1 = -5x^2 - 5x + 10 }

Moving all the terms to the left-hand side, we get:

5x2+10x−11=0{ 5x^2 + 10x - 11 = 0 }

This is a quadratic equation, and we can solve it using the quadratic formula:

x=−b±b2−4ac2a{ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }

In this case, a = 5, b = 10, and c = -11. Plugging these values into the formula, we get:

x=−10±100+22010{ x = \frac{-10 \pm \sqrt{100 + 220}}{10} }

Simplifying the expression under the square root, we get:

x=−10±32010{ x = \frac{-10 \pm \sqrt{320}}{10} }

Simplifying the square root, we get:

x=−10±8510{ x = \frac{-10 \pm 8\sqrt{5}}{10} }

Simplifying the expression, we get:

x=−5±455{ x = \frac{-5 \pm 4\sqrt{5}}{5} }

This is the solution to the equation.

Conclusion

In this article, we solved a complex equation involving fractions. We started by finding a common denominator for the fractions on the left-hand side, and then combined them into a single fraction. We then equated the fractions and solved for x using the quadratic formula. The solution to the equation is:

x=−5±455{ x = \frac{-5 \pm 4\sqrt{5}}{5} }

This is a quadratic equation, and it has two solutions. The solutions are:

x=−5+455{ x = \frac{-5 + 4\sqrt{5}}{5} }

and

x=−5−455{ x = \frac{-5 - 4\sqrt{5}}{5} }

These are the solutions to the equation.

Final Thoughts

Solving equations involving fractions can be challenging, but with a clear understanding of the steps involved, we can break them down and solve for the variable x. In this article, we used the quadratic formula to solve a complex equation, and we obtained two solutions. The solutions are:

x=−5+455{ x = \frac{-5 + 4\sqrt{5}}{5} }

and

x=−5−455{ x = \frac{-5 - 4\sqrt{5}}{5} }

These are the solutions to the equation.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Linear Algebra" by Jim Hefferon

Note: The references provided are for general information and are not specific to the equation solved in this article.

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Introduction

In our previous article, we solved a complex equation involving fractions. We started by finding a common denominator for the fractions on the left-hand side, and then combined them into a single fraction. We then equated the fractions and solved for x using the quadratic formula. In this article, we will answer some of the most frequently asked questions about solving equations involving fractions.

Q&A

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

A: The first step in solving an equation involving fractions is to find a common denominator for the fractions on the left-hand side. This will allow you to combine the fractions into a single fraction.

Q: How do I find a common denominator for fractions?

A: To find a common denominator for fractions, you need to identify the least common multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can divide into evenly.

Q: What is the quadratic formula?

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

x=−b±b2−4ac2a{ x = \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 to solve a quadratic equation, you need to plug in the values of a, b, and c into the formula. The values of a, b, and c are the coefficients of the quadratic equation.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. A quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I simplify a quadratic equation?

A: To simplify a quadratic equation, you need to combine like terms and factor out any common factors.

Q: What is the final step in solving an equation involving fractions?

A: The final step in solving an equation involving fractions is to check your solution by plugging it back into the original equation.

Common Mistakes to Avoid

When solving equations involving fractions, there are several common mistakes to avoid. These include:

  • Not finding a common denominator: Failing to find a common denominator can make it difficult to combine the fractions into a single fraction.
  • Not using the quadratic formula correctly: Using the quadratic formula incorrectly can lead to incorrect solutions.
  • Not checking the solution: Failing to check the solution can lead to incorrect answers.

Conclusion

Solving equations involving fractions can be challenging, but with a clear understanding of the steps involved, we can break them down and solve for the variable x. In this article, we answered some of the most frequently asked questions about solving equations involving fractions. We also discussed common mistakes to avoid and provided tips for simplifying quadratic equations.

Final Thoughts

Solving equations involving fractions requires patience, persistence, and practice. With these skills, you can tackle even the most complex equations and come out on top.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Linear Algebra" by Jim Hefferon

Note: The references provided are for general information and are not specific to the equation solved in this article.