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Questions and Answers
What is the first step to simplify the equation $ rac{5}{x - 1} + rac{1}{4 - 3x} = rac{3}{6x - 8}$?
If you multiply through by the common denominator, what will the equation look like?
What is the simplified form of the equation after collecting like terms and moving all terms to one side?
What method can be used to solve the quadratic equation $2x^2 - 7x + 3 = 0$?
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What are the approximate solutions for $x$ after solving the equation $2x^2 - 7x + 3 = 0$?
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Study Notes
Simplifying the Equation
- The first step to simplify the equation is to identify the common denominator of all the fractions involved.
- The common denominator for the fractions (\frac{5}{x - 1}), (\frac{1}{4 - 3x}), and (\frac{3}{6x - 8}) is ((x - 1)(4 - 3x)).
Multiplying Through by the Common Denominator
- Multiplying each term of the equation by the common denominator eliminates the fractions.
- After multiplying through, the equation becomes:
- (5(4 - 3x) + 1(x - 1) = 3(x - 1))
Collecting Like Terms and Rearranging
- Simplifying the resulting equation involves distributing and combining like terms.
- After rearranging all terms to one side, the simplified form is:
- (2x^2 - 7x + 3 = 0)
Solving the Quadratic Equation
- To solve the quadratic equation (2x^2 - 7x + 3 = 0), methods such as factoring, completing the square, or using the quadratic formula can be employed.
Approximate Solutions for (x)
- Applying the quadratic formula yields approximate solutions (x \approx 3.5) and (x \approx 0.4285), which can be rounded as necessary.
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Description
Test your skills on solving the equation $ rac{5}{x - 1} + rac{1}{4 - 3x} = rac{3}{6x - 8}$. This quiz will guide you through the steps of simplifying the expression, moving terms, and solving the quadratic equation $2x^2 - 7x + 3 = 0$. Find the approximate solutions and enhance your understanding of algebraic concepts.
