x+x^2=780

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Solution for x+x^2=780 equation:


Simplifying
x + x2 = 780

Solving
x + x2 = 780

Solving for variable 'x'.

Reorder the terms:
-780 + x + x2 = 780 + -780

Combine like terms: 780 + -780 = 0
-780 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '780' to each side of the equation.
-780 + x + 780 + x2 = 0 + 780

Reorder the terms:
-780 + 780 + x + x2 = 0 + 780

Combine like terms: -780 + 780 = 0
0 + x + x2 = 0 + 780
x + x2 = 0 + 780

Combine like terms: 0 + 780 = 780
x + x2 = 780

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x + 0.25 + x2 = 780 + 0.25

Reorder the terms:
0.25 + x + x2 = 780 + 0.25

Combine like terms: 780 + 0.25 = 780.25
0.25 + x + x2 = 780.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 780.25

Calculate the square root of the right side: 27.932955447

Break this problem into two subproblems by setting 
(x + 0.5) equal to 27.932955447 and -27.932955447.

Subproblem 1

x + 0.5 = 27.932955447 Simplifying x + 0.5 = 27.932955447 Reorder the terms: 0.5 + x = 27.932955447 Solving 0.5 + x = 27.932955447 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 27.932955447 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 27.932955447 + -0.5 x = 27.932955447 + -0.5 Combine like terms: 27.932955447 + -0.5 = 27.432955447 x = 27.432955447 Simplifying x = 27.432955447

Subproblem 2

x + 0.5 = -27.932955447 Simplifying x + 0.5 = -27.932955447 Reorder the terms: 0.5 + x = -27.932955447 Solving 0.5 + x = -27.932955447 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -27.932955447 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -27.932955447 + -0.5 x = -27.932955447 + -0.5 Combine like terms: -27.932955447 + -0.5 = -28.432955447 x = -28.432955447 Simplifying x = -28.432955447

Solution

The solution to the problem is based on the solutions from the subproblems. x = {27.432955447, -28.432955447}

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