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find an expression for $x$ in terms of $e$ when $e^2=\frac{x+3}{x-2}$

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Factor sums and differences of cubes

Full solution

Q. find an expression for

x

in terms of

e

when

e^2=\frac{x+3}{x-2}

Isolate fraction: Step $1$ : Start by isolating the fraction on one side: $\newline$ Given equation: $e^2 = \frac{x+3}{x-2}$ . $\newline$ Multiply both sides by $(x-2)$ to clear the fraction: $\newline$ $e^2 \cdot (x-2) = x + 3$ .
Expand left side: Step $2$ : Expand the left side: $e^2 \times x - 2e^2 = x + 3$ .
Bring $x$ terms together: Step $3$ : Bring all $x$ terms to one side: $\newline$ $e^2 \cdot x - x = 2e^2 + 3$ .
Factor out $x$ : Step $4$ : Factor out $x$ from the left side: $\newline$ $x \cdot (e^2 - 1) = 2e^2 + 3$ .
Solve for x: Step $5$ : Solve for x: $\newline$ $x = \frac{2e^2 + 3}{e^2 - 1}$ .

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