X^2+(x+1)^2-x(x+1)=57

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To solve this equation, we can first expand the expressions:

x^2 + (x+1)^2 - x(x+1) = 57

Expanding the binomial (x+1)^2:

x^2 + (x^2 + 2x + 1) - (x^2 + x) = 57

Now, simplify the equation:

x^2 + x^2 + 2x + 1 - x^2 - x = 57

Combine like terms:

2x^2 + 2x + 1 - x = 57

2x^2 + x + 1 = 57

Rearrange the terms to set the equation to zero:

2x^2 + x + 1 - 57 = 0

2x^2 + x - 56 = 0

Now, this is a quadratic equation that we can solve by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

x = [-1 ± sqrt(1 + 4256)] / (2*2)

x = [-1 ± sqrt(1 + 448)] / 4

x = [-1 ± sqrt(449)] / 4

x = [-1 ± 21] / 4

x = (20/4) or x = (-22/4)

Therefore, the solutions are x = 5 or x = -11.