Solve 2*(x-2)^2-3*(x-1)^2-(1-x)*(x-3)=0

$$2 \cdot ( x - 2 ) ^ { 2 } - 3 \cdot ( x - 1 ) ^ { 2 } - ( 1 - x ) \cdot ( x - 3 ) = 0$$

Solve for x

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

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

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

$$2x^{2}-8x+8-3\left(x-1\right)^{2}-\left(1-x\right)\left(x-3\right)=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$2x^{2}-8x+8-3\left(x^{2}-2x+1\right)-\left(1-x\right)\left(x-3\right)=0$$

Use the distributive property to multiply -3 by x^{2}-2x+1.

$$2x^{2}-8x+8-3x^{2}+6x-3-\left(1-x\right)\left(x-3\right)=0$$

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

$$-x^{2}-8x+8+6x-3-\left(1-x\right)\left(x-3\right)=0$$

Combine -8x and 6x to get -2x.

$$-x^{2}-2x+8-3-\left(1-x\right)\left(x-3\right)=0$$

Subtract 3 from 8 to get 5.

$$-x^{2}-2x+5-\left(1-x\right)\left(x-3\right)=0$$

Use the distributive property to multiply 1-x by x-3 and combine like terms.

$$-x^{2}-2x+5-\left(4x-3-x^{2}\right)=0$$

To find the opposite of 4x-3-x^{2}, find the opposite of each term.

$$-x^{2}-2x+5-4x+3+x^{2}=0$$

Combine -2x and -4x to get -6x.

$$-x^{2}-6x+5+3+x^{2}=0$$

Add 5 and 3 to get 8.

$$-x^{2}-6x+8+x^{2}=0$$

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

$$-6x+8=0$$

Subtract 8 from both sides. Anything subtracted from zero gives its negation.

$$-6x=-8$$

Divide both sides by -6.

$$x=\frac{-8}{-6}$$

Reduce the fraction \frac{-8}{-6} to lowest terms by extracting and canceling out -2.

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