Solve The Equation:${ \frac{5}{x+3}+\frac{4}{x+2}=2 }$Both Correct Answers Must Be Given To Gain The Marks.

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Introduction

In this article, we will delve into the world of algebra and solve a complex equation involving fractions. The equation is given as 5x+3+4x+2=2\frac{5}{x+3}+\frac{4}{x+2}=2. Our goal is to find the value of xx that satisfies this equation. We will break down the solution into manageable steps, making it easy to follow and understand.

Step 1: Multiply Both Sides by the Least Common Denominator (LCD)

To eliminate the fractions, we need to multiply both sides of the equation by the least common denominator (LCD). The LCD is the product of the denominators, which in this case is (x+3)(x+2)(x+3)(x+2). By multiplying both sides by the LCD, we get:

5(x+2)+4(x+3)=2(x+3)(x+2)5(x+2) + 4(x+3) = 2(x+3)(x+2)

Step 2: Expand and Simplify the Equation

Now, let's expand and simplify the equation by multiplying the terms:

5x+10+4x+12=2(x2+5x+6)5x + 10 + 4x + 12 = 2(x^2 + 5x + 6)

Combine like terms:

9x+22=2x2+10x+129x + 22 = 2x^2 + 10x + 12

Step 3: Rearrange the Equation

To make it easier to solve, let's rearrange the equation by moving all terms to one side:

2x2+10x+12−9x−22=02x^2 + 10x + 12 - 9x - 22 = 0

Combine like terms:

2x2+x−10=02x^2 + x - 10 = 0

Step 4: Solve the Quadratic Equation

We can solve this quadratic equation using the quadratic formula:

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

In this case, a=2a = 2, b=1b = 1, and c=−10c = -10. Plugging these values into the formula, we get:

x=−1±12−4(2)(−10)2(2)x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-10)}}{2(2)}

Simplify the expression under the square root:

x=−1±1+804x = \frac{-1 \pm \sqrt{1 + 80}}{4}

x=−1±814x = \frac{-1 \pm \sqrt{81}}{4}

x=−1±94x = \frac{-1 \pm 9}{4}

Step 5: Find the Two Possible Solutions

Now, let's find the two possible solutions by evaluating the expression for both the plus and minus signs:

x1=−1+94=84=2x_1 = \frac{-1 + 9}{4} = \frac{8}{4} = 2

x2=−1−94=−104=−52x_2 = \frac{-1 - 9}{4} = \frac{-10}{4} = -\frac{5}{2}

Conclusion

In this article, we solved the equation 5x+3+4x+2=2\frac{5}{x+3}+\frac{4}{x+2}=2 using a step-by-step approach. We multiplied both sides by the least common denominator, expanded and simplified the equation, rearranged it, and finally solved the quadratic equation using the quadratic formula. We found two possible solutions: x=2x = 2 and x=−52x = -\frac{5}{2}. Both solutions are correct and valid.

Final Answer

The final answer is 2\boxed{2} and −52\boxed{-\frac{5}{2}}.

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Introduction

In our previous article, we solved the equation 5x+3+4x+2=2\frac{5}{x+3}+\frac{4}{x+2}=2 using a step-by-step approach. We found two possible solutions: x=2x = 2 and x=−52x = -\frac{5}{2}. Both solutions are correct and valid. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into the solution.

Q&A

Q: What is the least common denominator (LCD) and why is it important?

A: The least common denominator (LCD) is the product of the denominators of the fractions in the equation. In this case, the LCD is (x+3)(x+2)(x+3)(x+2). It is important because multiplying both sides of the equation by the LCD eliminates the fractions and makes it easier to solve.

Q: How do I expand and simplify the equation?

A: To expand and simplify the equation, you need to multiply the terms and combine like terms. For example, in the equation 5x+10+4x+12=2(x+3)(x+2)5x + 10 + 4x + 12 = 2(x+3)(x+2), you would multiply the terms and combine like terms to get 9x+22=2x2+10x+129x + 22 = 2x^2 + 10x + 12.

Q: What is the quadratic formula and how do I use it?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

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

To use the quadratic formula, you need to identify the values of aa, bb, and cc in the equation. In this case, a=2a = 2, b=1b = 1, and c=−10c = -10. Plugging these values into the formula, you get:

x=−1±12−4(2)(−10)2(2)x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-10)}}{2(2)}

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

A: The two possible solutions to the equation are x=2x = 2 and x=−52x = -\frac{5}{2}. Both solutions are correct and valid.

Q: How do I check if the solutions are correct?

A: To check if the solutions are correct, you can plug them back into the original equation and see if they satisfy the equation. For example, if you plug x=2x = 2 into the original equation, you get:

52+3+42+2=2\frac{5}{2+3}+\frac{4}{2+2}=2

Simplifying the equation, you get:

55+44=2\frac{5}{5}+\frac{4}{4}=2

1+1=21+1=2

2=22=2

This shows that x=2x = 2 is a correct solution. Similarly, you can plug x=−52x = -\frac{5}{2} into the original equation and see if it satisfies the equation.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts and provide additional insights into the solution of the equation 5x+3+4x+2=2\frac{5}{x+3}+\frac{4}{x+2}=2. We answered questions about the least common denominator, expanding and simplifying the equation, the quadratic formula, and the two possible solutions. We also provided a step-by-step guide on how to check if the solutions are correct.

Final Answer

The final answer is 2\boxed{2} and −52\boxed{-\frac{5}{2}}.