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:
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.