3x(5+9x)=4x+13

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Solution for 3x(5+9x)=4x+13 equation:


Simplifying
3x(5 + 9x) = 4x + 13
(5 * 3x + 9x * 3x) = 4x + 13
(15x + 27x2) = 4x + 13

Reorder the terms:
15x + 27x2 = 13 + 4x

Solving
15x + 27x2 = 13 + 4x

Solving for variable 'x'.

Reorder the terms:
-13 + 15x + -4x + 27x2 = 13 + 4x + -13 + -4x

Combine like terms: 15x + -4x = 11x
-13 + 11x + 27x2 = 13 + 4x + -13 + -4x

Reorder the terms:
-13 + 11x + 27x2 = 13 + -13 + 4x + -4x

Combine like terms: 13 + -13 = 0
-13 + 11x + 27x2 = 0 + 4x + -4x
-13 + 11x + 27x2 = 4x + -4x

Combine like terms: 4x + -4x = 0
-13 + 11x + 27x2 = 0

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

Divide each side by '27'.
-0.4814814815 + 0.4074074074x + x2 = 0

Move the constant term to the right:

Add '0.4814814815' to each side of the equation.
-0.4814814815 + 0.4074074074x + 0.4814814815 + x2 = 0 + 0.4814814815

Reorder the terms:
-0.4814814815 + 0.4814814815 + 0.4074074074x + x2 = 0 + 0.4814814815

Combine like terms: -0.4814814815 + 0.4814814815 = 0.0000000000
0.0000000000 + 0.4074074074x + x2 = 0 + 0.4814814815
0.4074074074x + x2 = 0 + 0.4814814815

Combine like terms: 0 + 0.4814814815 = 0.4814814815
0.4074074074x + x2 = 0.4814814815

The x term is 0.4074074074x.  Take half its coefficient (0.2037037037).
Square it (0.04149519890) and add it to both sides.

Add '0.04149519890' to each side of the equation.
0.4074074074x + 0.04149519890 + x2 = 0.4814814815 + 0.04149519890

Reorder the terms:
0.04149519890 + 0.4074074074x + x2 = 0.4814814815 + 0.04149519890

Combine like terms: 0.4814814815 + 0.04149519890 = 0.5229766804
0.04149519890 + 0.4074074074x + x2 = 0.5229766804

Factor a perfect square on the left side:
(x + 0.2037037037)(x + 0.2037037037) = 0.5229766804

Calculate the square root of the right side: 0.723171266

Break this problem into two subproblems by setting 
(x + 0.2037037037) equal to 0.723171266 and -0.723171266.

Subproblem 1

x + 0.2037037037 = 0.723171266 Simplifying x + 0.2037037037 = 0.723171266 Reorder the terms: 0.2037037037 + x = 0.723171266 Solving 0.2037037037 + x = 0.723171266 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2037037037' to each side of the equation. 0.2037037037 + -0.2037037037 + x = 0.723171266 + -0.2037037037 Combine like terms: 0.2037037037 + -0.2037037037 = 0.0000000000 0.0000000000 + x = 0.723171266 + -0.2037037037 x = 0.723171266 + -0.2037037037 Combine like terms: 0.723171266 + -0.2037037037 = 0.5194675623 x = 0.5194675623 Simplifying x = 0.5194675623

Subproblem 2

x + 0.2037037037 = -0.723171266 Simplifying x + 0.2037037037 = -0.723171266 Reorder the terms: 0.2037037037 + x = -0.723171266 Solving 0.2037037037 + x = -0.723171266 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2037037037' to each side of the equation. 0.2037037037 + -0.2037037037 + x = -0.723171266 + -0.2037037037 Combine like terms: 0.2037037037 + -0.2037037037 = 0.0000000000 0.0000000000 + x = -0.723171266 + -0.2037037037 x = -0.723171266 + -0.2037037037 Combine like terms: -0.723171266 + -0.2037037037 = -0.9268749697 x = -0.9268749697 Simplifying x = -0.9268749697

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.5194675623, -0.9268749697}

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