x^2=x+7

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Solution for x^2=x+7 equation:


Simplifying
x2 = x + 7

Reorder the terms:
x2 = 7 + x

Solving
x2 = 7 + x

Solving for variable 'x'.

Reorder the terms:
-7 + -1x + x2 = 7 + x + -7 + -1x

Reorder the terms:
-7 + -1x + x2 = 7 + -7 + x + -1x

Combine like terms: 7 + -7 = 0
-7 + -1x + x2 = 0 + x + -1x
-7 + -1x + x2 = x + -1x

Combine like terms: x + -1x = 0
-7 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -1x + 7 + x2 = 0 + 7

Reorder the terms:
-7 + 7 + -1x + x2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -1x + x2 = 0 + 7
-1x + x2 = 0 + 7

Combine like terms: 0 + 7 = 7
-1x + x2 = 7

The x term is -1x.  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.
-1x + 0.25 + x2 = 7 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 7 + 0.25

Combine like terms: 7 + 0.25 = 7.25
0.25 + -1x + x2 = 7.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(x + -0.5) equal to 2.692582404 and -2.692582404.

Subproblem 1

x + -0.5 = 2.692582404 Simplifying x + -0.5 = 2.692582404 Reorder the terms: -0.5 + x = 2.692582404 Solving -0.5 + x = 2.692582404 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 = 2.692582404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 2.692582404 + 0.5 x = 2.692582404 + 0.5 Combine like terms: 2.692582404 + 0.5 = 3.192582404 x = 3.192582404 Simplifying x = 3.192582404

Subproblem 2

x + -0.5 = -2.692582404 Simplifying x + -0.5 = -2.692582404 Reorder the terms: -0.5 + x = -2.692582404 Solving -0.5 + x = -2.692582404 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 = -2.692582404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -2.692582404 + 0.5 x = -2.692582404 + 0.5 Combine like terms: -2.692582404 + 0.5 = -2.192582404 x = -2.192582404 Simplifying x = -2.192582404

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.192582404, -2.192582404}

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