5x(3x+3)=x+6

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Solution for 5x(3x+3)=x+6 equation:


Simplifying
5x(3x + 3) = x + 6

Reorder the terms:
5x(3 + 3x) = x + 6
(3 * 5x + 3x * 5x) = x + 6
(15x + 15x2) = x + 6

Reorder the terms:
15x + 15x2 = 6 + x

Solving
15x + 15x2 = 6 + x

Solving for variable 'x'.

Reorder the terms:
-6 + 15x + -1x + 15x2 = 6 + x + -6 + -1x

Combine like terms: 15x + -1x = 14x
-6 + 14x + 15x2 = 6 + x + -6 + -1x

Reorder the terms:
-6 + 14x + 15x2 = 6 + -6 + x + -1x

Combine like terms: 6 + -6 = 0
-6 + 14x + 15x2 = 0 + x + -1x
-6 + 14x + 15x2 = x + -1x

Combine like terms: x + -1x = 0
-6 + 14x + 15x2 = 0

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

Divide each side by '15'.
-0.4 + 0.9333333333x + x2 = 0

Move the constant term to the right:

Add '0.4' to each side of the equation.
-0.4 + 0.9333333333x + 0.4 + x2 = 0 + 0.4

Reorder the terms:
-0.4 + 0.4 + 0.9333333333x + x2 = 0 + 0.4

Combine like terms: -0.4 + 0.4 = 0.0
0.0 + 0.9333333333x + x2 = 0 + 0.4
0.9333333333x + x2 = 0 + 0.4

Combine like terms: 0 + 0.4 = 0.4
0.9333333333x + x2 = 0.4

The x term is 0.9333333333x.  Take half its coefficient (0.4666666667).
Square it (0.2177777778) and add it to both sides.

Add '0.2177777778' to each side of the equation.
0.9333333333x + 0.2177777778 + x2 = 0.4 + 0.2177777778

Reorder the terms:
0.2177777778 + 0.9333333333x + x2 = 0.4 + 0.2177777778

Combine like terms: 0.4 + 0.2177777778 = 0.6177777778
0.2177777778 + 0.9333333333x + x2 = 0.6177777778

Factor a perfect square on the left side:
(x + 0.4666666667)(x + 0.4666666667) = 0.6177777778

Calculate the square root of the right side: 0.785988408

Break this problem into two subproblems by setting 
(x + 0.4666666667) equal to 0.785988408 and -0.785988408.

Subproblem 1

x + 0.4666666667 = 0.785988408 Simplifying x + 0.4666666667 = 0.785988408 Reorder the terms: 0.4666666667 + x = 0.785988408 Solving 0.4666666667 + x = 0.785988408 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4666666667' to each side of the equation. 0.4666666667 + -0.4666666667 + x = 0.785988408 + -0.4666666667 Combine like terms: 0.4666666667 + -0.4666666667 = 0.0000000000 0.0000000000 + x = 0.785988408 + -0.4666666667 x = 0.785988408 + -0.4666666667 Combine like terms: 0.785988408 + -0.4666666667 = 0.3193217413 x = 0.3193217413 Simplifying x = 0.3193217413

Subproblem 2

x + 0.4666666667 = -0.785988408 Simplifying x + 0.4666666667 = -0.785988408 Reorder the terms: 0.4666666667 + x = -0.785988408 Solving 0.4666666667 + x = -0.785988408 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4666666667' to each side of the equation. 0.4666666667 + -0.4666666667 + x = -0.785988408 + -0.4666666667 Combine like terms: 0.4666666667 + -0.4666666667 = 0.0000000000 0.0000000000 + x = -0.785988408 + -0.4666666667 x = -0.785988408 + -0.4666666667 Combine like terms: -0.785988408 + -0.4666666667 = -1.2526550747 x = -1.2526550747 Simplifying x = -1.2526550747

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.3193217413, -1.2526550747}

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