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

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

Solve for x

$$x\neq \frac{3}{4}$$

Solution Steps

Variable x cannot be equal to \frac{3}{4} since division by zero is not defined. Multiply both sides of the equation by 4x-3.

$$\left(x^{2}-2x+2\right)\left(4x-3\right)^{-1}\left(4x-3\right)=x^{2}-2x+2$$

Use the distributive property to multiply x^{2}-2x+2 by \left(4x-3\right)^{-1}.

$$\left(x^{2}\left(4x-3\right)^{-1}-2x\left(4x-3\right)^{-1}+2\left(4x-3\right)^{-1}\right)\left(4x-3\right)=x^{2}-2x+2$$

Use the distributive property to multiply x^{2}\left(4x-3\right)^{-1}-2x\left(4x-3\right)^{-1}+2\left(4x-3\right)^{-1} by 4x-3 and combine like terms.

$$4\left(4x-3\right)^{-1}x^{3}-11x^{2}\left(4x-3\right)^{-1}+14x\left(4x-3\right)^{-1}-6\left(4x-3\right)^{-1}=x^{2}-2x+2$$

Subtract x^{2} from both sides.

$$4\left(4x-3\right)^{-1}x^{3}-11x^{2}\left(4x-3\right)^{-1}+14x\left(4x-3\right)^{-1}-6\left(4x-3\right)^{-1}-x^{2}=-2x+2$$

Add 2x to both sides.

$$4\left(4x-3\right)^{-1}x^{3}-11x^{2}\left(4x-3\right)^{-1}+14x\left(4x-3\right)^{-1}-6\left(4x-3\right)^{-1}-x^{2}+2x=2$$

Subtract 2 from both sides.

$$4\left(4x-3\right)^{-1}x^{3}-11x^{2}\left(4x-3\right)^{-1}+14x\left(4x-3\right)^{-1}-6\left(4x-3\right)^{-1}-x^{2}+2x-2=0$$

Reorder the terms.

$$4\times \frac{1}{4x-3}x^{3}-x^{2}-11\times \frac{1}{4x-3}x^{2}+2x+14\times \frac{1}{4x-3}x-2-6\times \frac{1}{4x-3}=0$$

Variable x cannot be equal to \frac{3}{4} since division by zero is not defined. Multiply both sides of the equation by 4x-3.

$$4\times 1x^{3}-x^{2}\left(4x-3\right)-11x^{2}+2x\left(4x-3\right)+14\times 1x+\left(4x-3\right)\left(-2\right)-6=0$$

Do the multiplications.

$$4x^{3}-x^{2}\left(4x-3\right)-11x^{2}+2x\left(4x-3\right)+14x+\left(4x-3\right)\left(-2\right)-6=0$$

Use the distributive property to multiply -x^{2} by 4x-3.

$$4x^{3}-4x^{3}+3x^{2}-11x^{2}+2x\left(4x-3\right)+14x+\left(4x-3\right)\left(-2\right)-6=0$$

Combine 4x^{3} and -4x^{3} to get 0.

$$3x^{2}-11x^{2}+2x\left(4x-3\right)+14x+\left(4x-3\right)\left(-2\right)-6=0$$

Combine 3x^{2} and -11x^{2} to get -8x^{2}.

$$-8x^{2}+2x\left(4x-3\right)+14x+\left(4x-3\right)\left(-2\right)-6=0$$

Use the distributive property to multiply 2x by 4x-3.

$$-8x^{2}+8x^{2}-6x+14x+\left(4x-3\right)\left(-2\right)-6=0$$

Combine -8x^{2} and 8x^{2} to get 0.

$$-6x+14x+\left(4x-3\right)\left(-2\right)-6=0$$

Combine -6x and 14x to get 8x.

$$8x+\left(4x-3\right)\left(-2\right)-6=0$$

Use the distributive property to multiply 4x-3 by -2.

$$8x-8x+6-6=0$$

Combine 8x and -8x to get 0.

$$6-6=0$$

Subtract 6 from 6 to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Variable x cannot be equal to \frac{3}{4}.

$$x\in \mathrm{R}\setminus \frac{3}{4}$$