Solve The Equation ${ \frac{2(x+7)}{x+4} - 2 = \frac{2x+20}{2x+8} }$

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Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand how to approach and solve various types of equations. In this article, we will focus on solving a specific equation involving fractions. We will break down the solution into manageable steps, making it easier to understand and follow along.

The Equation

The given equation is:

2(x+7)x+4−2=2x+202x+8\frac{2(x+7)}{x+4} - 2 = \frac{2x+20}{2x+8}

Our goal is to solve for the variable xx.

Step 1: Simplify the Left Side of the Equation

To simplify the left side of the equation, we can start by combining the terms:

2(x+7)x+4−2=2(x+7)x+4−2(x+4)x+4\frac{2(x+7)}{x+4} - 2 = \frac{2(x+7)}{x+4} - \frac{2(x+4)}{x+4}

This simplifies to:

2(x+7)−2(x+4)x+4=2x+14−2x−8x+4\frac{2(x+7) - 2(x+4)}{x+4} = \frac{2x+14 - 2x - 8}{x+4}

Simplifying further, we get:

6x+4=2x+202x+8\frac{6}{x+4} = \frac{2x+20}{2x+8}

Step 2: Eliminate the Fractions

To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is (x+4)(2x+8)(x+4)(2x+8):

(x+4)(2x+8)(6x+4)=(x+4)(2x+8)(2x+202x+8)(x+4)(2x+8)\left(\frac{6}{x+4}\right) = (x+4)(2x+8)\left(\frac{2x+20}{2x+8}\right)

This simplifies to:

6(2x+8)=(x+4)(2x+20)6(2x+8) = (x+4)(2x+20)

Step 3: Expand and Simplify

Expanding both sides of the equation, we get:

12x+48=2x2+24x+40x+8012x + 48 = 2x^2 + 24x + 40x + 80

Simplifying further, we get:

12x+48=2x2+64x+8012x + 48 = 2x^2 + 64x + 80

Step 4: Rearrange the Equation

Rearranging the equation to get all the terms on one side, we get:

2x2+52x+32=02x^2 + 52x + 32 = 0

Step 5: Solve the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, a=2a = 2, b=52b = 52, and c=32c = 32. Plugging these values into the formula, we get:

x=−52±522−4(2)(32)2(2)x = \frac{-52 \pm \sqrt{52^2 - 4(2)(32)}}{2(2)}

Simplifying further, we get:

x=−52±2704−2564x = \frac{-52 \pm \sqrt{2704 - 256}}{4}

x=−52±24484x = \frac{-52 \pm \sqrt{2448}}{4}

x=−52±49.64x = \frac{-52 \pm 49.6}{4}

Step 6: Find the Solutions

Simplifying further, we get two possible solutions:

x=−52+49.64=−2.44=−0.6x = \frac{-52 + 49.6}{4} = \frac{-2.4}{4} = -0.6

x=−52−49.64=−101.64=−25.4x = \frac{-52 - 49.6}{4} = \frac{-101.6}{4} = -25.4

Conclusion

In this article, we solved a specific equation involving fractions. We broke down the solution into manageable steps, making it easier to understand and follow along. We simplified the left side of the equation, eliminated the fractions, expanded and simplified, rearranged the equation, and finally solved the quadratic equation using the quadratic formula. The two possible solutions are x=−0.6x = -0.6 and x=−25.4x = -25.4.

Final Answer

The final answer is −0.6\boxed{-0.6} and −25.4\boxed{-25.4}.

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Introduction

In our previous article, we solved a specific equation involving fractions. We broke down the solution into manageable steps, making it easier to understand and follow along. In this article, we will provide a Q&A guide to help you better understand the solution and address any questions you may have.

Q: What is the main goal of solving the equation?

A: The main goal of solving the equation is to find the value of the variable xx.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to simplify the left side of the equation by combining the terms.

Q: How do we eliminate the fractions in the equation?

A: We can eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: What is the LCM of the denominators in this equation?

A: The LCM of the denominators is (x+4)(2x+8)(x+4)(2x+8).

Q: How do we expand and simplify the equation?

A: We expand and simplify the equation by multiplying out the terms and combining like terms.

Q: What is the final form of the equation after expanding and simplifying?

A: The final form of the equation after expanding and simplifying is 2x2+52x+32=02x^2 + 52x + 32 = 0.

Q: How do we solve the quadratic equation?

A: We can solve the quadratic equation using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Q: How do we plug in the values into the quadratic formula?

A: We plug in the values a=2a = 2, b=52b = 52, and c=32c = 32 into the quadratic formula.

Q: What are the two possible solutions to the equation?

A: The two possible solutions to the equation are x=−0.6x = -0.6 and x=−25.4x = -25.4.

Q: How do we determine which solution is correct?

A: We can determine which solution is correct by plugging the values back into the original equation and checking if the equation holds true.

Q: What is the final answer to the equation?

A: The final answer to the equation is −0.6\boxed{-0.6} and −25.4\boxed{-25.4}.

Common Mistakes to Avoid

When solving the equation, there are several common mistakes to avoid:

  • Not simplifying the left side of the equation
  • Not eliminating the fractions
  • Not expanding and simplifying the equation
  • Not plugging in the correct values into the quadratic formula
  • Not checking if the solutions are correct

Tips and Tricks

When solving the equation, here are some tips and tricks to keep in mind:

  • Make sure to simplify the left side of the equation before eliminating the fractions
  • Use the least common multiple (LCM) to eliminate the fractions
  • Expand and simplify the equation carefully to avoid mistakes
  • Plug in the correct values into the quadratic formula
  • Check if the solutions are correct by plugging the values back into the original equation

Conclusion

In this article, we provided a Q&A guide to help you better understand the solution to the equation and address any questions you may have. We covered the main goal of solving the equation, the first step in solving the equation, and how to eliminate the fractions. We also provided tips and tricks to keep in mind when solving the equation.