13x(x-6)=7(x-2)

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Solution for 13x(x-6)=7(x-2) equation:


Simplifying
13x(x + -6) = 7(x + -2)

Reorder the terms:
13x(-6 + x) = 7(x + -2)
(-6 * 13x + x * 13x) = 7(x + -2)
(-78x + 13x2) = 7(x + -2)

Reorder the terms:
-78x + 13x2 = 7(-2 + x)
-78x + 13x2 = (-2 * 7 + x * 7)
-78x + 13x2 = (-14 + 7x)

Solving
-78x + 13x2 = -14 + 7x

Solving for variable 'x'.

Reorder the terms:
14 + -78x + -7x + 13x2 = -14 + 7x + 14 + -7x

Combine like terms: -78x + -7x = -85x
14 + -85x + 13x2 = -14 + 7x + 14 + -7x

Reorder the terms:
14 + -85x + 13x2 = -14 + 14 + 7x + -7x

Combine like terms: -14 + 14 = 0
14 + -85x + 13x2 = 0 + 7x + -7x
14 + -85x + 13x2 = 7x + -7x

Combine like terms: 7x + -7x = 0
14 + -85x + 13x2 = 0

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

Divide each side by '13'.
1.076923077 + -6.538461538x + x2 = 0

Move the constant term to the right:

Add '-1.076923077' to each side of the equation.
1.076923077 + -6.538461538x + -1.076923077 + x2 = 0 + -1.076923077

Reorder the terms:
1.076923077 + -1.076923077 + -6.538461538x + x2 = 0 + -1.076923077

Combine like terms: 1.076923077 + -1.076923077 = 0.000000000
0.000000000 + -6.538461538x + x2 = 0 + -1.076923077
-6.538461538x + x2 = 0 + -1.076923077

Combine like terms: 0 + -1.076923077 = -1.076923077
-6.538461538x + x2 = -1.076923077

The x term is -6.538461538x.  Take half its coefficient (-3.269230769).
Square it (10.68786982) and add it to both sides.

Add '10.68786982' to each side of the equation.
-6.538461538x + 10.68786982 + x2 = -1.076923077 + 10.68786982

Reorder the terms:
10.68786982 + -6.538461538x + x2 = -1.076923077 + 10.68786982

Combine like terms: -1.076923077 + 10.68786982 = 9.610946743
10.68786982 + -6.538461538x + x2 = 9.610946743

Factor a perfect square on the left side:
(x + -3.269230769)(x + -3.269230769) = 9.610946743

Calculate the square root of the right side: 3.100152697

Break this problem into two subproblems by setting 
(x + -3.269230769) equal to 3.100152697 and -3.100152697.

Subproblem 1

x + -3.269230769 = 3.100152697 Simplifying x + -3.269230769 = 3.100152697 Reorder the terms: -3.269230769 + x = 3.100152697 Solving -3.269230769 + x = 3.100152697 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.269230769' to each side of the equation. -3.269230769 + 3.269230769 + x = 3.100152697 + 3.269230769 Combine like terms: -3.269230769 + 3.269230769 = 0.000000000 0.000000000 + x = 3.100152697 + 3.269230769 x = 3.100152697 + 3.269230769 Combine like terms: 3.100152697 + 3.269230769 = 6.369383466 x = 6.369383466 Simplifying x = 6.369383466

Subproblem 2

x + -3.269230769 = -3.100152697 Simplifying x + -3.269230769 = -3.100152697 Reorder the terms: -3.269230769 + x = -3.100152697 Solving -3.269230769 + x = -3.100152697 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.269230769' to each side of the equation. -3.269230769 + 3.269230769 + x = -3.100152697 + 3.269230769 Combine like terms: -3.269230769 + 3.269230769 = 0.000000000 0.000000000 + x = -3.100152697 + 3.269230769 x = -3.100152697 + 3.269230769 Combine like terms: -3.100152697 + 3.269230769 = 0.169078072 x = 0.169078072 Simplifying x = 0.169078072

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.369383466, 0.169078072}

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