Solve {l}{(3x-1)^{2}+17-3(2x+3)^{2}+42=2x(-3-x)-(x-1)^2}{(x^2-2(3x)(-1)+x^2)+17-3[2x^2-2(2x)(3)+3}

$$\left. \begin{array} { l } { ( 3 x - 1 ) ^ { 2 } + 17 - 3 ( 2 x + 3 ) ^ { 2 } + 42 = 2 x ( - 3 - x ) - ( x - 1 ) ^ { 2 } } \\ { ( x ^ { 2 } - 2 ( 3 x ) ( - 1 ) + x ^ { 2 } ) + 17 - 3 [ 2 x ^ { 2 } - 2 ( 2 x ) ( 3 ) + 3 } \end{array} \right.$$

Solve for x

$$x=\frac{17}{19}\approx 0.894736842$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-1\right)^{2}.

$$9x^{2}-6x+1+17-3\left(2x+3\right)^{2}+42=2x\left(-3-x\right)-\left(x-1\right)^{2}$$

Add 1 and 17 to get 18.

$$9x^{2}-6x+18-3\left(2x+3\right)^{2}+42=2x\left(-3-x\right)-\left(x-1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$9x^{2}-6x+18-3\left(4x^{2}+12x+9\right)+42=2x\left(-3-x\right)-\left(x-1\right)^{2}$$

Use the distributive property to multiply -3 by 4x^{2}+12x+9.

$$9x^{2}-6x+18-12x^{2}-36x-27+42=2x\left(-3-x\right)-\left(x-1\right)^{2}$$

Combine 9x^{2} and -12x^{2} to get -3x^{2}.

$$-3x^{2}-6x+18-36x-27+42=2x\left(-3-x\right)-\left(x-1\right)^{2}$$

Combine -6x and -36x to get -42x.

$$-3x^{2}-42x+18-27+42=2x\left(-3-x\right)-\left(x-1\right)^{2}$$

Subtract 27 from 18 to get -9.

$$-3x^{2}-42x-9+42=2x\left(-3-x\right)-\left(x-1\right)^{2}$$

Add -9 and 42 to get 33.

$$-3x^{2}-42x+33=2x\left(-3-x\right)-\left(x-1\right)^{2}$$

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

$$-3x^{2}-42x+33=-6x-2x^{2}-\left(x-1\right)^{2}$$

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

$$-3x^{2}-42x+33=-6x-2x^{2}-\left(x^{2}-2x+1\right)$$

To find the opposite of x^{2}-2x+1, find the opposite of each term.

$$-3x^{2}-42x+33=-6x-2x^{2}-x^{2}+2x-1$$

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

$$-3x^{2}-42x+33=-6x-3x^{2}+2x-1$$

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

$$-3x^{2}-42x+33=-4x-3x^{2}-1$$

Add 4x to both sides.

$$-3x^{2}-42x+33+4x=-3x^{2}-1$$

Combine -42x and 4x to get -38x.

$$-3x^{2}-38x+33=-3x^{2}-1$$

Add 3x^{2} to both sides.

$$-3x^{2}-38x+33+3x^{2}=-1$$

Combine -3x^{2} and 3x^{2} to get 0.

$$-38x+33=-1$$

Subtract 33 from both sides.

$$-38x=-1-33$$

Subtract 33 from -1 to get -34.

$$-38x=-34$$

Divide both sides by -38.

$$x=\frac{-34}{-38}$$

Reduce the fraction \frac{-34}{-38} to lowest terms by extracting and canceling out -2.

$$x=\frac{17}{19}$$