Solve The Equation: ${ \frac{3}{x-6} - \frac{5}{2x-8} = \frac{x-24}{x^2-10x+24} }$
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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 we will be solving is . This equation requires careful manipulation and simplification to arrive at the solution. We will break down the solution into manageable steps, making it easier to understand and follow along.
Step 1: Simplify the Equation
The first step in solving the equation is to simplify it by finding a common denominator for the fractions on the left-hand side. The common denominator is .
\frac{3}{x-6} - \frac{5}{2x-8} = \frac{x-24}{x^2-10x+24}
To simplify the equation, we need to multiply the numerator and denominator of each fraction by the necessary factors to obtain the common denominator.
\frac{3(2x-8)}{(x-6)(2x-8)} - \frac{5(x-6)}{(x-6)(2x-8)} = \frac{x-24}{x^2-10x+24}
Step 2: Combine the Fractions
Now that we have a common denominator, we can combine the fractions on the left-hand side.
\frac{3(2x-8) - 5(x-6)}{(x-6)(2x-8)} = \frac{x-24}{x^2-10x+24}
Step 3: Simplify the Numerator
Next, we need to simplify the numerator of the combined fraction.
\frac{6x-24 - 5x + 30}{(x-6)(2x-8)} = \frac{x-24}{x^2-10x+24}
Step 4: Factor the Numerator
We can factor the numerator to make it easier to work with.
\frac{x + 6}{(x-6)(2x-8)} = \frac{x-24}{x^2-10x+24}
Step 5: Cross-Multiply
To eliminate the fractions, we can cross-multiply.
(x + 6)(x^2-10x+24) = (x-24)(x-6)(2x-8)
Step 6: Expand and Simplify
We can expand and simplify both sides of the equation.
x^3 - 10x^2 + 24x + 6x^2 - 60x + 144 = (x-24)(2x^2 - 26x + 48)
Step 7: Simplify the Right-Hand Side
We can simplify the right-hand side of the equation.
x^3 - 4x^2 - 36x + 144 = 2x^3 - 52x^2 + 1152 - 26x^2 + 624x - 48x
Step 8: Combine Like Terms
We can combine like terms on both sides of the equation.
x^3 - 4x^2 - 36x + 144 = 2x^3 - 78x^2 + 624x - 48x
Step 9: Move All Terms to One Side
We can move all terms to one side of the equation to set it equal to zero.
x^3 - 4x^2 - 36x + 144 - 2x^3 + 78x^2 - 624x + 48x = 0
Step 10: Simplify the Equation
We can simplify the equation by combining like terms.
-x^3 + 74x^2 - 720x + 192 = 0
Step 11: Factor the Equation
We can factor the equation to make it easier to solve.
-(x - 8)(x^2 - 62x + 24) = 0
Step 12: Solve for x
We can solve for x by setting each factor equal to zero.
x - 8 = 0 \quad \text{or} \quad x^2 - 62x + 24 = 0
Step 13: Solve the Quadratic Equation
We can solve the quadratic equation using the quadratic formula.
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Step 14: Simplify the Solutions
We can simplify the solutions to obtain the final answer.
x = 8 \quad \text{or} \quad x = \frac{62 \pm \sqrt{3844 - 384}}{2}
Conclusion
In this article, we solved a complex equation involving fractions. We broke down the solution into manageable steps, making it easier to understand and follow along. We simplified the equation, combined the fractions, and factored the numerator. We then cross-multiplied, expanded and simplified, and combined like terms. Finally, we moved all terms to one side, simplified the equation, and factored it. We solved for x by setting each factor equal to zero and used the quadratic formula to solve the quadratic equation. The final answer is or .
Final Answer
The final answer is or .
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Introduction
In our previous article, we solved a complex equation involving fractions. We broke down the solution into manageable steps, making it easier to understand and follow along. In this article, we will answer some of the most frequently asked questions about solving the equation.
Q: What is the first step in solving the equation?
A: The first step in solving the equation is to simplify it by finding a common denominator for the fractions on the left-hand side.
Q: How do I find the common denominator?
A: To find the common denominator, we need to multiply the numerator and denominator of each fraction by the necessary factors to obtain the common denominator.
Q: What is the common denominator for the fractions on the left-hand side?
A: The common denominator for the fractions on the left-hand side is .
Q: How do I combine the fractions on the left-hand side?
A: To combine the fractions on the left-hand side, we need to multiply the numerator and denominator of each fraction by the necessary factors to obtain the common denominator.
Q: What is the next step in solving the equation?
A: The next step in solving the equation is to simplify the numerator of the combined fraction.
Q: How do I simplify the numerator?
A: To simplify the numerator, we can factor it to make it easier to work with.
Q: What is the next step in solving the equation?
A: The next step in solving the equation is to cross-multiply.
Q: How do I cross-multiply?
A: To cross-multiply, we need to multiply the numerator of the left-hand side fraction by the denominator of the right-hand side fraction and vice versa.
Q: What is the next step in solving the equation?
A: The next step in solving the equation is to expand and simplify both sides of the equation.
Q: How do I expand and simplify?
A: To expand and simplify, we need to multiply out the terms on both sides of the equation and combine like terms.
Q: What is the final step in solving the equation?
A: The final step in solving the equation is to move all terms to one side of the equation to set it equal to zero.
Q: How do I move all terms to one side of the equation?
A: To move all terms to one side of the equation, we need to add or subtract the necessary terms to make all the terms on one side of the equation.
Q: What is the final answer to the equation?
A: The final answer to the equation is or .
Conclusion
In this article, we answered some of the most frequently asked questions about solving the equation. We covered the first step in solving the equation, finding the common denominator, combining the fractions, simplifying the numerator, cross-multiplying, expanding and simplifying, and moving all terms to one side of the equation. We also provided the final answer to the equation.
Final Answer
The final answer is or .
Additional Resources
For more information on solving equations, please refer to the following resources:
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