51=x(10+0.19x)

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Solution for 51=x(10+0.19x) equation:


Simplifying
51 = x(10 + 0.19x)
51 = (10 * x + 0.19x * x)
51 = (10x + 0.19x2)

Solving
51 = 10x + 0.19x2

Solving for variable 'x'.

Reorder the terms:
51 + -10x + -0.19x2 = 10x + -10x + 0.19x2 + -0.19x2

Combine like terms: 10x + -10x = 0
51 + -10x + -0.19x2 = 0 + 0.19x2 + -0.19x2
51 + -10x + -0.19x2 = 0.19x2 + -0.19x2

Combine like terms: 0.19x2 + -0.19x2 = 0.00
51 + -10x + -0.19x2 = 0.00

Begin completing the square.  Divide all terms by
-0.19 the coefficient of the squared term: 

Divide each side by '-0.19'.
-268.4210526 + 52.63157895x + x2 = 0

Move the constant term to the right:

Add '268.4210526' to each side of the equation.
-268.4210526 + 52.63157895x + 268.4210526 + x2 = 0 + 268.4210526

Reorder the terms:
-268.4210526 + 268.4210526 + 52.63157895x + x2 = 0 + 268.4210526

Combine like terms: -268.4210526 + 268.4210526 = 0.0000000
0.0000000 + 52.63157895x + x2 = 0 + 268.4210526
52.63157895x + x2 = 0 + 268.4210526

Combine like terms: 0 + 268.4210526 = 268.4210526
52.63157895x + x2 = 268.4210526

The x term is 52.63157895x.  Take half its coefficient (26.31578948).
Square it (692.5207760) and add it to both sides.

Add '692.5207760' to each side of the equation.
52.63157895x + 692.5207760 + x2 = 268.4210526 + 692.5207760

Reorder the terms:
692.5207760 + 52.63157895x + x2 = 268.4210526 + 692.5207760

Combine like terms: 268.4210526 + 692.5207760 = 960.9418286
692.5207760 + 52.63157895x + x2 = 960.9418286

Factor a perfect square on the left side:
(x + 26.31578948)(x + 26.31578948) = 960.9418286

Calculate the square root of the right side: 30.999061737

Break this problem into two subproblems by setting 
(x + 26.31578948) equal to 30.999061737 and -30.999061737.

Subproblem 1

x + 26.31578948 = 30.999061737 Simplifying x + 26.31578948 = 30.999061737 Reorder the terms: 26.31578948 + x = 30.999061737 Solving 26.31578948 + x = 30.999061737 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-26.31578948' to each side of the equation. 26.31578948 + -26.31578948 + x = 30.999061737 + -26.31578948 Combine like terms: 26.31578948 + -26.31578948 = 0.00000000 0.00000000 + x = 30.999061737 + -26.31578948 x = 30.999061737 + -26.31578948 Combine like terms: 30.999061737 + -26.31578948 = 4.683272257 x = 4.683272257 Simplifying x = 4.683272257

Subproblem 2

x + 26.31578948 = -30.999061737 Simplifying x + 26.31578948 = -30.999061737 Reorder the terms: 26.31578948 + x = -30.999061737 Solving 26.31578948 + x = -30.999061737 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-26.31578948' to each side of the equation. 26.31578948 + -26.31578948 + x = -30.999061737 + -26.31578948 Combine like terms: 26.31578948 + -26.31578948 = 0.00000000 0.00000000 + x = -30.999061737 + -26.31578948 x = -30.999061737 + -26.31578948 Combine like terms: -30.999061737 + -26.31578948 = -57.314851217 x = -57.314851217 Simplifying x = -57.314851217

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.683272257, -57.314851217}

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