3x(x+5)+4x=78

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Solution for 3x(x+5)+4x=78 equation:


Simplifying
3x(x + 5) + 4x = 78

Reorder the terms:
3x(5 + x) + 4x = 78
(5 * 3x + x * 3x) + 4x = 78
(15x + 3x2) + 4x = 78

Reorder the terms:
15x + 4x + 3x2 = 78

Combine like terms: 15x + 4x = 19x
19x + 3x2 = 78

Solving
19x + 3x2 = 78

Solving for variable 'x'.

Reorder the terms:
-78 + 19x + 3x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-78 + 19x + 3x2 = 0

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

Divide each side by '3'.
-26 + 6.333333333x + x2 = 0

Move the constant term to the right:

Add '26' to each side of the equation.
-26 + 6.333333333x + 26 + x2 = 0 + 26

Reorder the terms:
-26 + 26 + 6.333333333x + x2 = 0 + 26

Combine like terms: -26 + 26 = 0
0 + 6.333333333x + x2 = 0 + 26
6.333333333x + x2 = 0 + 26

Combine like terms: 0 + 26 = 26
6.333333333x + x2 = 26

The x term is 6.333333333x.  Take half its coefficient (3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
6.333333333x + 10.02777778 + x2 = 26 + 10.02777778

Reorder the terms:
10.02777778 + 6.333333333x + x2 = 26 + 10.02777778

Combine like terms: 26 + 10.02777778 = 36.02777778
10.02777778 + 6.333333333x + x2 = 36.02777778

Factor a perfect square on the left side:
(x + 3.166666667)(x + 3.166666667) = 36.02777778

Calculate the square root of the right side: 6.002314369

Break this problem into two subproblems by setting 
(x + 3.166666667) equal to 6.002314369 and -6.002314369.

Subproblem 1

x + 3.166666667 = 6.002314369 Simplifying x + 3.166666667 = 6.002314369 Reorder the terms: 3.166666667 + x = 6.002314369 Solving 3.166666667 + x = 6.002314369 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.166666667' to each side of the equation. 3.166666667 + -3.166666667 + x = 6.002314369 + -3.166666667 Combine like terms: 3.166666667 + -3.166666667 = 0.000000000 0.000000000 + x = 6.002314369 + -3.166666667 x = 6.002314369 + -3.166666667 Combine like terms: 6.002314369 + -3.166666667 = 2.835647702 x = 2.835647702 Simplifying x = 2.835647702

Subproblem 2

x + 3.166666667 = -6.002314369 Simplifying x + 3.166666667 = -6.002314369 Reorder the terms: 3.166666667 + x = -6.002314369 Solving 3.166666667 + x = -6.002314369 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.166666667' to each side of the equation. 3.166666667 + -3.166666667 + x = -6.002314369 + -3.166666667 Combine like terms: 3.166666667 + -3.166666667 = 0.000000000 0.000000000 + x = -6.002314369 + -3.166666667 x = -6.002314369 + -3.166666667 Combine like terms: -6.002314369 + -3.166666667 = -9.168981036 x = -9.168981036 Simplifying x = -9.168981036

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.835647702, -9.168981036}

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