x(5x+1)=1

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


Simplifying
x(5x + 1) = 1

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

Solving
1x + 5x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + 1x + 5x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 1x + 5x2 = 0

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

Divide each side by '5'.
-0.2 + 0.2x + x2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + 0.2x + 0.2 + x2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + 0.2x + x2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + 0.2x + x2 = 0 + 0.2
0.2x + x2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
0.2x + x2 = 0.2

The x term is 0.2x.  Take half its coefficient (0.1).
Square it (0.01) and add it to both sides.

Add '0.01' to each side of the equation.
0.2x + 0.01 + x2 = 0.2 + 0.01

Reorder the terms:
0.01 + 0.2x + x2 = 0.2 + 0.01

Combine like terms: 0.2 + 0.01 = 0.21
0.01 + 0.2x + x2 = 0.21

Factor a perfect square on the left side:
(x + 0.1)(x + 0.1) = 0.21

Calculate the square root of the right side: 0.458257569

Break this problem into two subproblems by setting 
(x + 0.1) equal to 0.458257569 and -0.458257569.

Subproblem 1

x + 0.1 = 0.458257569 Simplifying x + 0.1 = 0.458257569 Reorder the terms: 0.1 + x = 0.458257569 Solving 0.1 + x = 0.458257569 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = 0.458257569 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = 0.458257569 + -0.1 x = 0.458257569 + -0.1 Combine like terms: 0.458257569 + -0.1 = 0.358257569 x = 0.358257569 Simplifying x = 0.358257569

Subproblem 2

x + 0.1 = -0.458257569 Simplifying x + 0.1 = -0.458257569 Reorder the terms: 0.1 + x = -0.458257569 Solving 0.1 + x = -0.458257569 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = -0.458257569 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = -0.458257569 + -0.1 x = -0.458257569 + -0.1 Combine like terms: -0.458257569 + -0.1 = -0.558257569 x = -0.558257569 Simplifying x = -0.558257569

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.358257569, -0.558257569}

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