3x(4+x)+5(x-3)=38

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Solution for 3x(4+x)+5(x-3)=38 equation:


Simplifying
3x(4 + x) + 5(x + -3) = 38
(4 * 3x + x * 3x) + 5(x + -3) = 38
(12x + 3x2) + 5(x + -3) = 38

Reorder the terms:
12x + 3x2 + 5(-3 + x) = 38
12x + 3x2 + (-3 * 5 + x * 5) = 38
12x + 3x2 + (-15 + 5x) = 38

Reorder the terms:
-15 + 12x + 5x + 3x2 = 38

Combine like terms: 12x + 5x = 17x
-15 + 17x + 3x2 = 38

Solving
-15 + 17x + 3x2 = 38

Solving for variable 'x'.

Reorder the terms:
-15 + -38 + 17x + 3x2 = 38 + -38

Combine like terms: -15 + -38 = -53
-53 + 17x + 3x2 = 38 + -38

Combine like terms: 38 + -38 = 0
-53 + 17x + 3x2 = 0

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

Divide each side by '3'.
-17.66666667 + 5.666666667x + x2 = 0

Move the constant term to the right:

Add '17.66666667' to each side of the equation.
-17.66666667 + 5.666666667x + 17.66666667 + x2 = 0 + 17.66666667

Reorder the terms:
-17.66666667 + 17.66666667 + 5.666666667x + x2 = 0 + 17.66666667

Combine like terms: -17.66666667 + 17.66666667 = 0.00000000
0.00000000 + 5.666666667x + x2 = 0 + 17.66666667
5.666666667x + x2 = 0 + 17.66666667

Combine like terms: 0 + 17.66666667 = 17.66666667
5.666666667x + x2 = 17.66666667

The x term is 5.666666667x.  Take half its coefficient (2.833333334).
Square it (8.027777782) and add it to both sides.

Add '8.027777782' to each side of the equation.
5.666666667x + 8.027777782 + x2 = 17.66666667 + 8.027777782

Reorder the terms:
8.027777782 + 5.666666667x + x2 = 17.66666667 + 8.027777782

Combine like terms: 17.66666667 + 8.027777782 = 25.694444452
8.027777782 + 5.666666667x + x2 = 25.694444452

Factor a perfect square on the left side:
(x + 2.833333334)(x + 2.833333334) = 25.694444452

Calculate the square root of the right side: 5.068968776

Break this problem into two subproblems by setting 
(x + 2.833333334) equal to 5.068968776 and -5.068968776.

Subproblem 1

x + 2.833333334 = 5.068968776 Simplifying x + 2.833333334 = 5.068968776 Reorder the terms: 2.833333334 + x = 5.068968776 Solving 2.833333334 + x = 5.068968776 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.833333334' to each side of the equation. 2.833333334 + -2.833333334 + x = 5.068968776 + -2.833333334 Combine like terms: 2.833333334 + -2.833333334 = 0.000000000 0.000000000 + x = 5.068968776 + -2.833333334 x = 5.068968776 + -2.833333334 Combine like terms: 5.068968776 + -2.833333334 = 2.235635442 x = 2.235635442 Simplifying x = 2.235635442

Subproblem 2

x + 2.833333334 = -5.068968776 Simplifying x + 2.833333334 = -5.068968776 Reorder the terms: 2.833333334 + x = -5.068968776 Solving 2.833333334 + x = -5.068968776 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.833333334' to each side of the equation. 2.833333334 + -2.833333334 + x = -5.068968776 + -2.833333334 Combine like terms: 2.833333334 + -2.833333334 = 0.000000000 0.000000000 + x = -5.068968776 + -2.833333334 x = -5.068968776 + -2.833333334 Combine like terms: -5.068968776 + -2.833333334 = -7.90230211 x = -7.90230211 Simplifying x = -7.90230211

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.235635442, -7.90230211}

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