x(x+1)=224

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Solution for x(x+1)=224 equation:


Simplifying
x(x + 1) = 224

Reorder the terms:
x(1 + x) = 224
(1 * x + x * x) = 224
(1x + x2) = 224

Solving
1x + x2 = 224

Solving for variable 'x'.

Reorder the terms:
-224 + 1x + x2 = 224 + -224

Combine like terms: 224 + -224 = 0
-224 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '224' to each side of the equation.
-224 + 1x + 224 + x2 = 0 + 224

Reorder the terms:
-224 + 224 + 1x + x2 = 0 + 224

Combine like terms: -224 + 224 = 0
0 + 1x + x2 = 0 + 224
1x + x2 = 0 + 224

Combine like terms: 0 + 224 = 224
1x + x2 = 224

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 = 224 + 0.25

Reorder the terms:
0.25 + x + x2 = 224 + 0.25

Combine like terms: 224 + 0.25 = 224.25
0.25 + x + x2 = 224.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 224.25

Calculate the square root of the right side: 14.974979132

Break this problem into two subproblems by setting 
(x + 0.5) equal to 14.974979132 and -14.974979132.

Subproblem 1

x + 0.5 = 14.974979132 Simplifying x + 0.5 = 14.974979132 Reorder the terms: 0.5 + x = 14.974979132 Solving 0.5 + x = 14.974979132 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 = 14.974979132 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 14.974979132 + -0.5 x = 14.974979132 + -0.5 Combine like terms: 14.974979132 + -0.5 = 14.474979132 x = 14.474979132 Simplifying x = 14.474979132

Subproblem 2

x + 0.5 = -14.974979132 Simplifying x + 0.5 = -14.974979132 Reorder the terms: 0.5 + x = -14.974979132 Solving 0.5 + x = -14.974979132 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 = -14.974979132 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -14.974979132 + -0.5 x = -14.974979132 + -0.5 Combine like terms: -14.974979132 + -0.5 = -15.474979132 x = -15.474979132 Simplifying x = -15.474979132

Solution

The solution to the problem is based on the solutions from the subproblems. x = {14.474979132, -15.474979132}

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