10x(3x-13)=40

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Solution for 10x(3x-13)=40 equation: 


Simplifying
10x(3x + -13) = 40

Reorder the terms:
10x(-13 + 3x) = 40
(-13 * 10x + 3x * 10x) = 40
(-130x + 30x2) = 40

Solving
-130x + 30x2 = 40

Solving for variable 'x'.

Reorder the terms:
-40 + -130x + 30x2 = 40 + -40

Combine like terms: 40 + -40 = 0
-40 + -130x + 30x2 = 0

Factor out the Greatest Common Factor (GCF), '10'.
10(-4 + -13x + 3x2) = 0

Ignore the factor 10.

Subproblem 1
Set the factor '(-4 + -13x + 3x2)' equal to zero and attempt to solve:

Simplifying
-4 + -13x + 3x2 = 0

Solving
-4 + -13x + 3x2 = 0

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

Divide each side by '3'.
-1.333333333 + -4.333333333x + x2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + -4.333333333x + 1.333333333 + x2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + -4.333333333x + x2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + -4.333333333x + x2 = 0 + 1.333333333
-4.333333333x + x2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
-4.333333333x + x2 = 1.333333333

The x term is -4.333333333x.  Take half its coefficient (-2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
-4.333333333x + 4.694444446 + x2 = 1.333333333 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333x + x2 = 1.333333333 + 4.694444446

Combine like terms: 1.333333333 + 4.694444446 = 6.027777779
4.694444446 + -4.333333333x + x2 = 6.027777779

Factor a perfect square on the left side:
(x + -2.166666667)(x + -2.166666667) = 6.027777779

Calculate the square root of the right side: 2.455153311

Break this problem into two subproblems by setting 
(x + -2.166666667) equal to 2.455153311 and -2.455153311.


Subproblem 1
x + -2.166666667 = 2.455153311

Simplifying
x + -2.166666667 = 2.455153311

Reorder the terms:
-2.166666667 + x = 2.455153311

Solving
-2.166666667 + x = 2.455153311

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = 2.455153311 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = 2.455153311 + 2.166666667
x = 2.455153311 + 2.166666667

Combine like terms: 2.455153311 + 2.166666667 = 4.621819978
x = 4.621819978

Simplifying
x = 4.621819978


Subproblem 2
x + -2.166666667 = -2.455153311

Simplifying
x + -2.166666667 = -2.455153311

Reorder the terms:
-2.166666667 + x = -2.455153311

Solving
-2.166666667 + x = -2.455153311

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = -2.455153311 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = -2.455153311 + 2.166666667
x = -2.455153311 + 2.166666667

Combine like terms: -2.455153311 + 2.166666667 = -0.288486644
x = -0.288486644

Simplifying
x = -0.288486644


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.621819978, -0.288486644}
Solution
x = {4.621819978, -0.288486644}

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