Solve for x: (24x^2 + 25x - 47) / (9x-2) = 8x - 3 - 53 / (9x-2)
Understand the Problem
The question requires solving the equation: $\frac{24x^2 + 25x - 47}{9x-2} = 8x - 3 - \frac{53}{9x-2}$. The problem involves algebraic manipulations to isolate x and find its value.
Answer
$x = 0, \frac{17}{12}$
Answer for screen readers
$x = 0, \frac{17}{12}$
Steps to Solve
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Add the fractions Add $\frac{53}{9x-2}$ to both sides of the equation to combine the fractions with the same denominator: $$ \frac{24x^2 + 25x - 47}{9x-2} + \frac{53}{9x-2} = 8x - 3 $$
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Combine like terms Combine the fractions on the left side: $$ \frac{24x^2 + 25x - 47 + 53}{9x-2} = 8x - 3 $$ Simplify the numerator: $$ \frac{24x^2 + 25x + 6}{9x-2} = 8x - 3 $$
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Multiply both sides by $9x-2$ Multiply both sides to eliminate the denominator: $$ 24x^2 + 25x + 6 = (8x - 3)(9x - 2) $$
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Expand the right side Expand the right side of the equation: $$ 24x^2 + 25x + 6 = 72x^2 - 16x - 27x + 6 $$ Simplify the right side: $$ 24x^2 + 25x + 6 = 72x^2 - 43x + 6 $$
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Rearrange the equation Move all terms to one side to set the equation to zero: $$ 0 = 72x^2 - 24x^2 - 43x - 25x + 6 - 6 $$ Combine like terms: $$ 0 = 48x^2 - 68x $$
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Factor out a common factor Factor out $4x$ from the equation: $$ 0 = 4x(12x - 17) $$
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Solve for x Set each factor equal to zero and solve for $x$: $4x = 0$ or $12x - 17 = 0$ $x = 0$ or $12x = 17$ $x = 0$ or $x = \frac{17}{12}$
$x = 0, \frac{17}{12}$
More Information
Both $x = 0$ and $x = \frac{17}{12}$ are valid solutions to the original equation.
Tips
A common mistake is to forget to distribute the terms correctly when expanding $(8x - 3)(9x - 2)$. Another common mistake is to only solve for one of the possible values of x after factoring. Also students may forget that $x = \frac{2}{9}$ is not in the domain of the function, however this value was not obtained as a solution.
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