x(x+1)=89

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Solution for x(x+1)=89 equation:


Simplifying
x(x + 1) = 89

Reorder the terms:
x(1 + x) = 89
(1 * x + x * x) = 89
(1x + x2) = 89

Solving
1x + x2 = 89

Solving for variable 'x'.

Reorder the terms:
-89 + 1x + x2 = 89 + -89

Combine like terms: 89 + -89 = 0
-89 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '89' to each side of the equation.
-89 + 1x + 89 + x2 = 0 + 89

Reorder the terms:
-89 + 89 + 1x + x2 = 0 + 89

Combine like terms: -89 + 89 = 0
0 + 1x + x2 = 0 + 89
1x + x2 = 0 + 89

Combine like terms: 0 + 89 = 89
1x + x2 = 89

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 = 89 + 0.25

Reorder the terms:
0.25 + x + x2 = 89 + 0.25

Combine like terms: 89 + 0.25 = 89.25
0.25 + x + x2 = 89.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 89.25

Calculate the square root of the right side: 9.447221814

Break this problem into two subproblems by setting 
(x + 0.5) equal to 9.447221814 and -9.447221814.

Subproblem 1

x + 0.5 = 9.447221814 Simplifying x + 0.5 = 9.447221814 Reorder the terms: 0.5 + x = 9.447221814 Solving 0.5 + x = 9.447221814 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 = 9.447221814 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 9.447221814 + -0.5 x = 9.447221814 + -0.5 Combine like terms: 9.447221814 + -0.5 = 8.947221814 x = 8.947221814 Simplifying x = 8.947221814

Subproblem 2

x + 0.5 = -9.447221814 Simplifying x + 0.5 = -9.447221814 Reorder the terms: 0.5 + x = -9.447221814 Solving 0.5 + x = -9.447221814 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 = -9.447221814 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -9.447221814 + -0.5 x = -9.447221814 + -0.5 Combine like terms: -9.447221814 + -0.5 = -9.947221814 x = -9.947221814 Simplifying x = -9.947221814

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.947221814, -9.947221814}

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