x(x+1)=75

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Solution for x(x+1)=75 equation:


Simplifying
x(x + 1) = 75

Reorder the terms:
x(1 + x) = 75
(1 * x + x * x) = 75
(1x + x2) = 75

Solving
1x + x2 = 75

Solving for variable 'x'.

Reorder the terms:
-75 + 1x + x2 = 75 + -75

Combine like terms: 75 + -75 = 0
-75 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '75' to each side of the equation.
-75 + 1x + 75 + x2 = 0 + 75

Reorder the terms:
-75 + 75 + 1x + x2 = 0 + 75

Combine like terms: -75 + 75 = 0
0 + 1x + x2 = 0 + 75
1x + x2 = 0 + 75

Combine like terms: 0 + 75 = 75
1x + x2 = 75

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 = 75 + 0.25

Reorder the terms:
0.25 + x + x2 = 75 + 0.25

Combine like terms: 75 + 0.25 = 75.25
0.25 + x + x2 = 75.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 75.25

Calculate the square root of the right side: 8.674675786

Break this problem into two subproblems by setting 
(x + 0.5) equal to 8.674675786 and -8.674675786.

Subproblem 1

x + 0.5 = 8.674675786 Simplifying x + 0.5 = 8.674675786 Reorder the terms: 0.5 + x = 8.674675786 Solving 0.5 + x = 8.674675786 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 = 8.674675786 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.674675786 + -0.5 x = 8.674675786 + -0.5 Combine like terms: 8.674675786 + -0.5 = 8.174675786 x = 8.174675786 Simplifying x = 8.174675786

Subproblem 2

x + 0.5 = -8.674675786 Simplifying x + 0.5 = -8.674675786 Reorder the terms: 0.5 + x = -8.674675786 Solving 0.5 + x = -8.674675786 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 = -8.674675786 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.674675786 + -0.5 x = -8.674675786 + -0.5 Combine like terms: -8.674675786 + -0.5 = -9.174675786 x = -9.174675786 Simplifying x = -9.174675786

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.174675786, -9.174675786}

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