3x(1+5x)=0.5x+14

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Solution for 3x(1+5x)=0.5x+14 equation:


Simplifying
3x(1 + 5x) = 0.5x + 14
(1 * 3x + 5x * 3x) = 0.5x + 14
(3x + 15x2) = 0.5x + 14

Reorder the terms:
3x + 15x2 = 14 + 0.5x

Solving
3x + 15x2 = 14 + 0.5x

Solving for variable 'x'.

Reorder the terms:
-14 + 3x + -0.5x + 15x2 = 14 + 0.5x + -14 + -0.5x

Combine like terms: 3x + -0.5x = 2.5x
-14 + 2.5x + 15x2 = 14 + 0.5x + -14 + -0.5x

Reorder the terms:
-14 + 2.5x + 15x2 = 14 + -14 + 0.5x + -0.5x

Combine like terms: 14 + -14 = 0
-14 + 2.5x + 15x2 = 0 + 0.5x + -0.5x
-14 + 2.5x + 15x2 = 0.5x + -0.5x

Combine like terms: 0.5x + -0.5x = 0.0
-14 + 2.5x + 15x2 = 0.0

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

Divide each side by '15'.
-0.9333333333 + 0.1666666667x + x2 = 0.0

Move the constant term to the right:

Add '0.9333333333' to each side of the equation.
-0.9333333333 + 0.1666666667x + 0.9333333333 + x2 = 0.0 + 0.9333333333

Reorder the terms:
-0.9333333333 + 0.9333333333 + 0.1666666667x + x2 = 0.0 + 0.9333333333

Combine like terms: -0.9333333333 + 0.9333333333 = 0.0000000000
0.0000000000 + 0.1666666667x + x2 = 0.0 + 0.9333333333
0.1666666667x + x2 = 0.0 + 0.9333333333

Combine like terms: 0.0 + 0.9333333333 = 0.9333333333
0.1666666667x + x2 = 0.9333333333

The x term is 0.1666666667x.  Take half its coefficient (0.08333333335).
Square it (0.006944444447) and add it to both sides.

Add '0.006944444447' to each side of the equation.
0.1666666667x + 0.006944444447 + x2 = 0.9333333333 + 0.006944444447

Reorder the terms:
0.006944444447 + 0.1666666667x + x2 = 0.9333333333 + 0.006944444447

Combine like terms: 0.9333333333 + 0.006944444447 = 0.940277777747
0.006944444447 + 0.1666666667x + x2 = 0.940277777747

Factor a perfect square on the left side:
(x + 0.08333333335)(x + 0.08333333335) = 0.940277777747

Calculate the square root of the right side: 0.969679214

Break this problem into two subproblems by setting 
(x + 0.08333333335) equal to 0.969679214 and -0.969679214.

Subproblem 1

x + 0.08333333335 = 0.969679214 Simplifying x + 0.08333333335 = 0.969679214 Reorder the terms: 0.08333333335 + x = 0.969679214 Solving 0.08333333335 + x = 0.969679214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = 0.969679214 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = 0.969679214 + -0.08333333335 x = 0.969679214 + -0.08333333335 Combine like terms: 0.969679214 + -0.08333333335 = 0.88634588065 x = 0.88634588065 Simplifying x = 0.88634588065

Subproblem 2

x + 0.08333333335 = -0.969679214 Simplifying x + 0.08333333335 = -0.969679214 Reorder the terms: 0.08333333335 + x = -0.969679214 Solving 0.08333333335 + x = -0.969679214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = -0.969679214 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = -0.969679214 + -0.08333333335 x = -0.969679214 + -0.08333333335 Combine like terms: -0.969679214 + -0.08333333335 = -1.05301254735 x = -1.05301254735 Simplifying x = -1.05301254735

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.88634588065, -1.05301254735}

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