x(x+1)=54

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Solution for x(x+1)=54 equation:


Simplifying
x(x + 1) = 54

Reorder the terms:
x(1 + x) = 54
(1 * x + x * x) = 54
(1x + x2) = 54

Solving
1x + x2 = 54

Solving for variable 'x'.

Reorder the terms:
-54 + 1x + x2 = 54 + -54

Combine like terms: 54 + -54 = 0
-54 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + 1x + 54 + x2 = 0 + 54

Reorder the terms:
-54 + 54 + 1x + x2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + 1x + x2 = 0 + 54
1x + x2 = 0 + 54

Combine like terms: 0 + 54 = 54
1x + x2 = 54

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 = 54 + 0.25

Reorder the terms:
0.25 + x + x2 = 54 + 0.25

Combine like terms: 54 + 0.25 = 54.25
0.25 + x + x2 = 54.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 54.25

Calculate the square root of the right side: 7.365459931

Break this problem into two subproblems by setting 
(x + 0.5) equal to 7.365459931 and -7.365459931.

Subproblem 1

x + 0.5 = 7.365459931 Simplifying x + 0.5 = 7.365459931 Reorder the terms: 0.5 + x = 7.365459931 Solving 0.5 + x = 7.365459931 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 = 7.365459931 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 7.365459931 + -0.5 x = 7.365459931 + -0.5 Combine like terms: 7.365459931 + -0.5 = 6.865459931 x = 6.865459931 Simplifying x = 6.865459931

Subproblem 2

x + 0.5 = -7.365459931 Simplifying x + 0.5 = -7.365459931 Reorder the terms: 0.5 + x = -7.365459931 Solving 0.5 + x = -7.365459931 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 = -7.365459931 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -7.365459931 + -0.5 x = -7.365459931 + -0.5 Combine like terms: -7.365459931 + -0.5 = -7.865459931 x = -7.865459931 Simplifying x = -7.865459931

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.865459931, -7.865459931}

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