5x(x+1)=22

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Solution for 5x(x+1)=22 equation:


Simplifying
5x(x + 1) = 22

Reorder the terms:
5x(1 + x) = 22
(1 * 5x + x * 5x) = 22
(5x + 5x2) = 22

Solving
5x + 5x2 = 22

Solving for variable 'x'.

Reorder the terms:
-22 + 5x + 5x2 = 22 + -22

Combine like terms: 22 + -22 = 0
-22 + 5x + 5x2 = 0

Begin completing the square.  Divide all terms by
5 the coefficient of the squared term: 

Divide each side by '5'.
-4.4 + x + x2 = 0

Move the constant term to the right:

Add '4.4' to each side of the equation.
-4.4 + x + 4.4 + x2 = 0 + 4.4

Reorder the terms:
-4.4 + 4.4 + x + x2 = 0 + 4.4

Combine like terms: -4.4 + 4.4 = 0.0
0.0 + x + x2 = 0 + 4.4
x + x2 = 0 + 4.4

Combine like terms: 0 + 4.4 = 4.4
x + x2 = 4.4

The x term is x.  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.
 + 0.25 + x2 = 4.4 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 4.4 + 0.25

Combine like terms: 4.4 + 0.25 = 4.65
1.25 + x2 = 4.65

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 4.65

Calculate the square root of the right side: 2.156385865

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.156385865 and -2.156385865.

Subproblem 1

x + 0.5 = 2.156385865 Simplifying x + 0.5 = 2.156385865 Reorder the terms: 0.5 + x = 2.156385865 Solving 0.5 + x = 2.156385865 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 = 2.156385865 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 2.156385865 + -0.5 x = 2.156385865 + -0.5 Combine like terms: 2.156385865 + -0.5 = 1.656385865 x = 1.656385865 Simplifying x = 1.656385865

Subproblem 2

x + 0.5 = -2.156385865 Simplifying x + 0.5 = -2.156385865 Reorder the terms: 0.5 + x = -2.156385865 Solving 0.5 + x = -2.156385865 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 = -2.156385865 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -2.156385865 + -0.5 x = -2.156385865 + -0.5 Combine like terms: -2.156385865 + -0.5 = -2.656385865 x = -2.656385865 Simplifying x = -2.656385865

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.656385865, -2.656385865}

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