x^2=x+21

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Solution for x^2=x+21 equation:


Simplifying
x2 = x + 21

Reorder the terms:
x2 = 21 + x

Solving
x2 = 21 + x

Solving for variable 'x'.

Reorder the terms:
-21 + -1x + x2 = 21 + x + -21 + -1x

Reorder the terms:
-21 + -1x + x2 = 21 + -21 + x + -1x

Combine like terms: 21 + -21 = 0
-21 + -1x + x2 = 0 + x + -1x
-21 + -1x + x2 = x + -1x

Combine like terms: x + -1x = 0
-21 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + -1x + 21 + x2 = 0 + 21

Reorder the terms:
-21 + 21 + -1x + x2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + -1x + x2 = 0 + 21
-1x + x2 = 0 + 21

Combine like terms: 0 + 21 = 21
-1x + x2 = 21

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 = 21 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 21 + 0.25

Combine like terms: 21 + 0.25 = 21.25
0.25 + -1x + x2 = 21.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(x + -0.5) equal to 4.609772229 and -4.609772229.

Subproblem 1

x + -0.5 = 4.609772229 Simplifying x + -0.5 = 4.609772229 Reorder the terms: -0.5 + x = 4.609772229 Solving -0.5 + x = 4.609772229 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 = 4.609772229 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 4.609772229 + 0.5 x = 4.609772229 + 0.5 Combine like terms: 4.609772229 + 0.5 = 5.109772229 x = 5.109772229 Simplifying x = 5.109772229

Subproblem 2

x + -0.5 = -4.609772229 Simplifying x + -0.5 = -4.609772229 Reorder the terms: -0.5 + x = -4.609772229 Solving -0.5 + x = -4.609772229 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 = -4.609772229 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -4.609772229 + 0.5 x = -4.609772229 + 0.5 Combine like terms: -4.609772229 + 0.5 = -4.109772229 x = -4.109772229 Simplifying x = -4.109772229

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.109772229, -4.109772229}

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