3x(4+x)=24+x

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Solution for 3x(4+x)=24+x equation:


Simplifying
3x(4 + x) = 24 + x
(4 * 3x + x * 3x) = 24 + x
(12x + 3x2) = 24 + x

Solving
12x + 3x2 = 24 + x

Solving for variable 'x'.

Reorder the terms:
-24 + 12x + -1x + 3x2 = 24 + x + -24 + -1x

Combine like terms: 12x + -1x = 11x
-24 + 11x + 3x2 = 24 + x + -24 + -1x

Reorder the terms:
-24 + 11x + 3x2 = 24 + -24 + x + -1x

Combine like terms: 24 + -24 = 0
-24 + 11x + 3x2 = 0 + x + -1x
-24 + 11x + 3x2 = x + -1x

Combine like terms: x + -1x = 0
-24 + 11x + 3x2 = 0

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

Divide each side by '3'.
-8 + 3.666666667x + x2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 3.666666667x + 8 + x2 = 0 + 8

Reorder the terms:
-8 + 8 + 3.666666667x + x2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 3.666666667x + x2 = 0 + 8
3.666666667x + x2 = 0 + 8

Combine like terms: 0 + 8 = 8
3.666666667x + x2 = 8

The x term is 3.666666667x.  Take half its coefficient (1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
3.666666667x + 3.361111114 + x2 = 8 + 3.361111114

Reorder the terms:
3.361111114 + 3.666666667x + x2 = 8 + 3.361111114

Combine like terms: 8 + 3.361111114 = 11.361111114
3.361111114 + 3.666666667x + x2 = 11.361111114

Factor a perfect square on the left side:
(x + 1.833333334)(x + 1.833333334) = 11.361111114

Calculate the square root of the right side: 3.370624736

Break this problem into two subproblems by setting 
(x + 1.833333334) equal to 3.370624736 and -3.370624736.

Subproblem 1

x + 1.833333334 = 3.370624736 Simplifying x + 1.833333334 = 3.370624736 Reorder the terms: 1.833333334 + x = 3.370624736 Solving 1.833333334 + x = 3.370624736 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = 3.370624736 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = 3.370624736 + -1.833333334 x = 3.370624736 + -1.833333334 Combine like terms: 3.370624736 + -1.833333334 = 1.537291402 x = 1.537291402 Simplifying x = 1.537291402

Subproblem 2

x + 1.833333334 = -3.370624736 Simplifying x + 1.833333334 = -3.370624736 Reorder the terms: 1.833333334 + x = -3.370624736 Solving 1.833333334 + x = -3.370624736 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = -3.370624736 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = -3.370624736 + -1.833333334 x = -3.370624736 + -1.833333334 Combine like terms: -3.370624736 + -1.833333334 = -5.20395807 x = -5.20395807 Simplifying x = -5.20395807

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.537291402, -5.20395807}

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