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# Algebra

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Find all real values of x such that

(2x^2 - 3x)/(x^2 - x) + 5x - 11 = (3x^2 + 4)/(x^2 - 1)

Jun 1, 2021

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$$\frac{2x^{2}-3x}{x^{2}-x}+5x-11=\frac{3x^{2}+4}{x^{2}-1}$$ (Problem)

$$\frac{x(2x-3)}{x(x-1)}+5x-11=\frac{3x^{2}+4}{(x+1)(x-1)}$$  (Factor out the x and expand $$x^{2}-1$$)

$$\frac{2x-3}{x-1}+5x-11=\frac{3x^{2}+4}{(x+1)(x-1)}$$   (Cancel the x)

$$(2x-3)(x+1)+5x(x-1)(x+1)-11(x-1)(x+1)=3x^{2}+4$$  (Multiply everything by (x-1)(x+1) to get rid of the denominator on the right hand side)

$$(2x^{2}-x-3)+(5x^{3}-5x)-(11x^{2}-11)=3x^{2}+4$$  (Expand)

$$5x^{3}-9x^{2}-6x+8=3x^{2}+4$$   (Simplify)

$$5x^{3}-12x^{2}-6x-4=0$$   (Simplify)

I hope you can take it from here!

Jun 2, 2021