Solve the Equation: x²+(x-2)²=2(x+1)² with Square Binomials

Solve the following problem:

$x^2+(x-2)^2=2(x+1)^2$

Examine the given equation:

$x^2+(x-2)^2=2(x+1)^2$

Start by simplifying the equation, to achieve this we'll apply the perfect square formula for a binomial squared:

$(a\pm b)^2=a^2\pm2ab+b^2$,

Start by opening the parentheses on both sides simultaneously using the perfect square formula, then proceed to combine like terms,

Note that according to the order of operations (which prioritizes exponents over multiplication), the expression inside of the right-hand parentheses is first squared and then the resulting expression is multiplied by 2,

Therefore, the expression that we obtain from applying the perfect square formula on the right-hand side will be placed in parentheses which we'll multiply by 2 (highlighted with an underline in the following calculation):

$x^2+(x-2)^2=2\underline{(x+1)^2} \\ \downarrow\\ x^2+x^2-2\cdot x\cdot2+2^2=2\underline{(x^2+2\cdot x\cdot1+1^2)}\\ x^2+x^2-4x+4=2(x^2+2x+1)\\$ Let's continue, open the parentheses on the right side by using the distributive property, move and combine like terms. In the final step we'll solve the simplified equation that we obtain

$x^2+x^2-4x+4=2(x^2+2x+1)\\ 2x^2-4x+4=2x^2+4x+2\\ -8x=-2\hspace{6pt}\text{/:(-8)}\\ x=\frac{-2}{-8}\\ \downarrow\\ \boxed{x=\frac{1}{4}}$

In the final step we reduced the fraction that we obtained as the solution for the unknown,

Therefore the correct answer is answer B.