3x(5-x)=1

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Solution for 3x(5-x)=1 equation:


Simplifying
3x(5 + -1x) = 1
(5 * 3x + -1x * 3x) = 1
(15x + -3x2) = 1

Solving
15x + -3x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + 15x + -3x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 15x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
0.3333333333 + -5x + x2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -5x + -0.3333333333 + x2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -5x + x2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -5x + x2 = 0 + -0.3333333333
-5x + x2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-5x + x2 = -0.3333333333

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = -0.3333333333 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -0.3333333333 + 6.25

Combine like terms: -0.3333333333 + 6.25 = 5.9166666667
6.25 + -5x + x2 = 5.9166666667

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 5.9166666667

Calculate the square root of the right side: 2.43241992

Break this problem into two subproblems by setting 
(x + -2.5) equal to 2.43241992 and -2.43241992.

Subproblem 1

x + -2.5 = 2.43241992 Simplifying x + -2.5 = 2.43241992 Reorder the terms: -2.5 + x = 2.43241992 Solving -2.5 + x = 2.43241992 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 2.43241992 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 2.43241992 + 2.5 x = 2.43241992 + 2.5 Combine like terms: 2.43241992 + 2.5 = 4.93241992 x = 4.93241992 Simplifying x = 4.93241992

Subproblem 2

x + -2.5 = -2.43241992 Simplifying x + -2.5 = -2.43241992 Reorder the terms: -2.5 + x = -2.43241992 Solving -2.5 + x = -2.43241992 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -2.43241992 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -2.43241992 + 2.5 x = -2.43241992 + 2.5 Combine like terms: -2.43241992 + 2.5 = 0.06758008 x = 0.06758008 Simplifying x = 0.06758008

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.93241992, 0.06758008}

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