Solve The Equation:${ \frac{1}{x+3} + \frac{5x+10}{x} = 3 }$Provide Both Correct Answers To Gain Full Marks.

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Introduction

Solving equations is a fundamental concept in mathematics, and it requires a deep understanding of algebraic manipulations. In this article, we will focus on solving a specific equation involving fractions, and we will provide both correct answers to gain full marks. The equation is given as:

1x+3+5x+10x=3\frac{1}{x+3} + \frac{5x+10}{x} = 3

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is x(x+3)x(x+3). Multiplying both sides by x(x+3)x(x+3), we get:

x(x+3)(1x+3+5x+10x)=3x(x+3)x(x+3)\left(\frac{1}{x+3} + \frac{5x+10}{x}\right) = 3x(x+3)

Step 2: Distribute and Simplify

Distributing the x(x+3)x(x+3) on the left-hand side, we get:

x(xx+3)+(x+3)(5x+10x)=3x(x+3)x\left(\frac{x}{x+3}\right) + (x+3)\left(\frac{5x+10}{x}\right) = 3x(x+3)

Simplifying the fractions, we get:

x2x+3+(x+3)(5x+10)x=3x(x+3)\frac{x^2}{x+3} + \frac{(x+3)(5x+10)}{x} = 3x(x+3)

Step 3: Eliminate the Fractions

To eliminate the fractions, we can multiply both sides of the equation by the LCM of the denominators, which is x(x+3)x(x+3). Multiplying both sides by x(x+3)x(x+3), we get:

x2+(x+3)(5x+10)=3x(x+3)2x^2 + (x+3)(5x+10) = 3x(x+3)^2

Step 4: Expand and Simplify

Expanding the left-hand side, we get:

x2+5x2+10x+15x+30=3x(x+3)2x^2 + 5x^2 + 10x + 15x + 30 = 3x(x+3)^2

Simplifying the left-hand side, we get:

6x2+25x+30=3x(x+3)26x^2 + 25x + 30 = 3x(x+3)^2

Step 5: Expand the Right-Hand Side

Expanding the right-hand side, we get:

3x(x2+6x+9)=3x3+18x2+27x3x(x^2 + 6x + 9) = 3x^3 + 18x^2 + 27x

Step 6: Equate the Left-Hand Side and the Right-Hand Side

Equating the left-hand side and the right-hand side, we get:

6x2+25x+30=3x3+18x2+27x6x^2 + 25x + 30 = 3x^3 + 18x^2 + 27x

Step 7: Rearrange the Terms

Rearranging the terms, we get:

3x3−12x2−2x−30=03x^3 - 12x^2 - 2x - 30 = 0

Step 8: Factor the Cubic Equation

Factoring the cubic equation, we get:

(x−5)(3x2+3x+6)=0(x-5)(3x^2 + 3x + 6) = 0

Step 9: Solve for x

Solving for x, we get:

x−5=0⇒x=5x-5 = 0 \Rightarrow x = 5

or

3x2+3x+6=03x^2 + 3x + 6 = 0

Using the quadratic formula, we get:

x=−3±32−4(3)(6)2(3)x = \frac{-3 \pm \sqrt{3^2 - 4(3)(6)}}{2(3)}

Simplifying the expression, we get:

x=−3±−636x = \frac{-3 \pm \sqrt{-63}}{6}

Conclusion

In this article, we have solved the equation 1x+3+5x+10x=3\frac{1}{x+3} + \frac{5x+10}{x} = 3 using a step-by-step approach. We have provided both correct answers to gain full marks. The first answer is x=5x = 5, and the second answer is x=−3±−636x = \frac{-3 \pm \sqrt{-63}}{6}.

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Introduction

In our previous article, we solved the equation 1x+3+5x+10x=3\frac{1}{x+3} + \frac{5x+10}{x} = 3 using a step-by-step approach. In this article, we will provide a Q&A guide to help you understand the solution better.

Q: What is the least common multiple (LCM) of the denominators?

A: The LCM of the denominators is x(x+3)x(x+3).

Q: Why do we need to multiply both sides of the equation by the LCM?

A: We need to multiply both sides of the equation by the LCM to eliminate the fractions.

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

A: The first step in solving the equation is to multiply both sides of the equation by the LCM, which is x(x+3)x(x+3).

Q: How do we simplify the left-hand side of the equation?

A: We simplify the left-hand side of the equation by distributing the x(x+3)x(x+3) on the left-hand side.

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

A: The second step in solving the equation is to eliminate the fractions by multiplying both sides of the equation by the LCM.

Q: How do we expand the right-hand side of the equation?

A: We expand the right-hand side of the equation by multiplying the 3x3x by the (x+3)2(x+3)^2.

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

A: The third step in solving the equation is to equate the left-hand side and the right-hand side of the equation.

Q: How do we rearrange the terms in the equation?

A: We rearrange the terms in the equation by moving all the terms to one side of the equation.

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

A: The fourth step in solving the equation is to factor the cubic equation.

Q: How do we solve for x?

A: We solve for x by setting each factor equal to zero and solving for x.

Q: What are the two possible solutions for x?

A: The two possible solutions for x are x=5x = 5 and x=−3±−636x = \frac{-3 \pm \sqrt{-63}}{6}.

Q: Why do we need to use the quadratic formula to solve for x?

A: We need to use the quadratic formula to solve for x because the equation is a quadratic equation.

Q: What is the significance of the quadratic formula?

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

Q: Can you explain the concept of the least common multiple (LCM)?

A: The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the given numbers.

Q: How do you find the LCM of two or more numbers?

A: To find the LCM of two or more numbers, you can list the multiples of each number and find the smallest number that is common to all the lists.

Q: What is the difference between the LCM and the greatest common divisor (GCD)?

A: The LCM is the smallest number that is a multiple of each of the given numbers, while the GCD is the largest number that divides each of the given numbers.

Conclusion

In this article, we have provided a Q&A guide to help you understand the solution to the equation 1x+3+5x+10x=3\frac{1}{x+3} + \frac{5x+10}{x} = 3. We hope that this guide has been helpful in clarifying any doubts you may have had about the solution.