x(x+1)=22

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Solution for x(x+1)=22 equation:


Simplifying
x(x + 1) = 22

Reorder the terms:
x(1 + x) = 22
(1 * x + x * x) = 22
(1x + x2) = 22

Solving
1x + x2 = 22

Solving for variable 'x'.

Reorder the terms:
-22 + 1x + x2 = 22 + -22

Combine like terms: 22 + -22 = 0
-22 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '22' to each side of the equation.
-22 + 1x + 22 + x2 = 0 + 22

Reorder the terms:
-22 + 22 + 1x + x2 = 0 + 22

Combine like terms: -22 + 22 = 0
0 + 1x + x2 = 0 + 22
1x + x2 = 0 + 22

Combine like terms: 0 + 22 = 22
1x + x2 = 22

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.
x + 0.25 + x2 = 22 + 0.25

Reorder the terms:
0.25 + x + x2 = 22 + 0.25

Combine like terms: 22 + 0.25 = 22.25
0.25 + x + x2 = 22.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 22.25

Calculate the square root of the right side: 4.716990566

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.716990566 and -4.716990566.

Subproblem 1

x + 0.5 = 4.716990566 Simplifying x + 0.5 = 4.716990566 Reorder the terms: 0.5 + x = 4.716990566 Solving 0.5 + x = 4.716990566 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 = 4.716990566 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 4.716990566 + -0.5 x = 4.716990566 + -0.5 Combine like terms: 4.716990566 + -0.5 = 4.216990566 x = 4.216990566 Simplifying x = 4.216990566

Subproblem 2

x + 0.5 = -4.716990566 Simplifying x + 0.5 = -4.716990566 Reorder the terms: 0.5 + x = -4.716990566 Solving 0.5 + x = -4.716990566 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 = -4.716990566 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -4.716990566 + -0.5 x = -4.716990566 + -0.5 Combine like terms: -4.716990566 + -0.5 = -5.216990566 x = -5.216990566 Simplifying x = -5.216990566

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.216990566, -5.216990566}

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