Solve (4x-4)(x-1)/2+4x-2+(11-9x^2)=(3-2x)(2x+3)+3x(5-x)-15x+2

$$\frac { ( 4 x - 4 ) ( x - 1 ) } { 2 } + 4 x - 2 + ( 11 - 9 x ^ { 2 } ) = ( 3 - 2 x ) ( 2 x + 3 ) + 3 x ( 5 - x ) - 15 x + 2$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

Multiply both sides of the equation by 2.

$$\left(4x-4\right)\left(x-1\right)+8x-4+22-18x^{2}=2\left(3-2x\right)\left(2x+3\right)+6x\left(5-x\right)-30x+4$$

Use the distributive property to multiply 4x-4 by x-1 and combine like terms.

$$4x^{2}-8x+4+8x-4+22-18x^{2}=2\left(3-2x\right)\left(2x+3\right)+6x\left(5-x\right)-30x+4$$

Combine -8x and 8x to get 0.

$$4x^{2}+4-4+22-18x^{2}=2\left(3-2x\right)\left(2x+3\right)+6x\left(5-x\right)-30x+4$$

Subtract 4 from 4 to get 0.

$$4x^{2}+22-18x^{2}=2\left(3-2x\right)\left(2x+3\right)+6x\left(5-x\right)-30x+4$$

Combine 4x^{2} and -18x^{2} to get -14x^{2}.

$$-14x^{2}+22=2\left(3-2x\right)\left(2x+3\right)+6x\left(5-x\right)-30x+4$$

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

$$-14x^{2}+22=\left(6-4x\right)\left(2x+3\right)+6x\left(5-x\right)-30x+4$$

Use the distributive property to multiply 6-4x by 2x+3 and combine like terms.

$$-14x^{2}+22=18-8x^{2}+6x\left(5-x\right)-30x+4$$

Use the distributive property to multiply 6x by 5-x.

$$-14x^{2}+22=18-8x^{2}+30x-6x^{2}-30x+4$$

Combine -8x^{2} and -6x^{2} to get -14x^{2}.

$$-14x^{2}+22=18-14x^{2}+30x-30x+4$$

Combine 30x and -30x to get 0.

$$-14x^{2}+22=18-14x^{2}+4$$

Add 18 and 4 to get 22.

$$-14x^{2}+22=22-14x^{2}$$

Subtract 22 from both sides.

$$-14x^{2}+22-22=-14x^{2}$$

Subtract 22 from 22 to get 0.

$$-14x^{2}=-14x^{2}$$

Add 14x^{2} to both sides.

$$-14x^{2}+14x^{2}=0$$

Combine -14x^{2} and 14x^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for x

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

Solution Steps

Multiply both sides of the equation by 2.

$$\left(4x-4\right)\left(x-1\right)+8x-4+22-18x^{2}=2\left(3-2x\right)\left(2x+3\right)+6x\left(5-x\right)-30x+4$$

Use the distributive property to multiply 4x-4 by x-1 and combine like terms.

$$4x^{2}-8x+4+8x-4+22-18x^{2}=2\left(3-2x\right)\left(2x+3\right)+6x\left(5-x\right)-30x+4$$

Combine -8x and 8x to get 0.

$$4x^{2}+4-4+22-18x^{2}=2\left(3-2x\right)\left(2x+3\right)+6x\left(5-x\right)-30x+4$$

Subtract 4 from 4 to get 0.

$$4x^{2}+22-18x^{2}=2\left(3-2x\right)\left(2x+3\right)+6x\left(5-x\right)-30x+4$$

Combine 4x^{2} and -18x^{2} to get -14x^{2}.

$$-14x^{2}+22=2\left(3-2x\right)\left(2x+3\right)+6x\left(5-x\right)-30x+4$$

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

$$-14x^{2}+22=\left(6-4x\right)\left(2x+3\right)+6x\left(5-x\right)-30x+4$$

Use the distributive property to multiply 6-4x by 2x+3 and combine like terms.

$$-14x^{2}+22=18-8x^{2}+6x\left(5-x\right)-30x+4$$

Use the distributive property to multiply 6x by 5-x.

$$-14x^{2}+22=18-8x^{2}+30x-6x^{2}-30x+4$$

Combine -8x^{2} and -6x^{2} to get -14x^{2}.

$$-14x^{2}+22=18-14x^{2}+30x-30x+4$$

Combine 30x and -30x to get 0.

$$-14x^{2}+22=18-14x^{2}+4$$

Add 18 and 4 to get 22.

$$-14x^{2}+22=22-14x^{2}$$

Subtract 22 from both sides.

$$-14x^{2}+22-22=-14x^{2}$$

Subtract 22 from 22 to get 0.

$$-14x^{2}=-14x^{2}$$

Add 14x^{2} to both sides.

$$-14x^{2}+14x^{2}=0$$

Combine -14x^{2} and 14x^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

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