Solve (2-x)(2+x)+(2x-5)^2=3(1-x)(5-x)+6

$$( 2 - x ) ( 2 + x ) + ( 2 x - 5 ) ^ { 2 } = 3 ( 1 - x ) ( 5 - x ) + 6$$

Solve for x

$$x=4$$

Steps for Solving Linear Equation

Consider \left(2-x\right)\left(2+x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-5\right)^{2}.

$$4-x^{2}+4x^{2}-20x+25=3\left(1-x\right)\left(5-x\right)+6$$

Combine -x^{2} and 4x^{2} to get 3x^{2}.

$$4+3x^{2}-20x+25=3\left(1-x\right)\left(5-x\right)+6$$

Add 4 and 25 to get 29.

$$29+3x^{2}-20x=3\left(1-x\right)\left(5-x\right)+6$$

Use the distributive property to multiply 3 by 1-x.

$$29+3x^{2}-20x=\left(3-3x\right)\left(5-x\right)+6$$

Use the distributive property to multiply 3-3x by 5-x and combine like terms.

$$29+3x^{2}-20x=15-18x+3x^{2}+6$$

Add 15 and 6 to get 21.

$$29+3x^{2}-20x=21-18x+3x^{2}$$

Add 18x to both sides.

$$29+3x^{2}-20x+18x=21+3x^{2}$$

Combine -20x and 18x to get -2x.

$$29+3x^{2}-2x=21+3x^{2}$$

Subtract 3x^{2} from both sides.

$$29+3x^{2}-2x-3x^{2}=21$$

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

$$29-2x=21$$

Subtract 29 from both sides.

$$-2x=21-29$$

Subtract 29 from 21 to get -8.

$$-2x=-8$$

Divide both sides by -2.

$$x=\frac{-8}{-2}$$

Divide -8 by -2 to get 4.

$$x=4$$