3x(x-3)+13=2(x+5)

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Solution for 3x(x-3)+13=2(x+5) equation:


Simplifying
3x(x + -3) + 13 = 2(x + 5)

Reorder the terms:
3x(-3 + x) + 13 = 2(x + 5)
(-3 * 3x + x * 3x) + 13 = 2(x + 5)
(-9x + 3x2) + 13 = 2(x + 5)

Reorder the terms:
13 + -9x + 3x2 = 2(x + 5)

Reorder the terms:
13 + -9x + 3x2 = 2(5 + x)
13 + -9x + 3x2 = (5 * 2 + x * 2)
13 + -9x + 3x2 = (10 + 2x)

Solving
13 + -9x + 3x2 = 10 + 2x

Solving for variable 'x'.

Reorder the terms:
13 + -10 + -9x + -2x + 3x2 = 10 + 2x + -10 + -2x

Combine like terms: 13 + -10 = 3
3 + -9x + -2x + 3x2 = 10 + 2x + -10 + -2x

Combine like terms: -9x + -2x = -11x
3 + -11x + 3x2 = 10 + 2x + -10 + -2x

Reorder the terms:
3 + -11x + 3x2 = 10 + -10 + 2x + -2x

Combine like terms: 10 + -10 = 0
3 + -11x + 3x2 = 0 + 2x + -2x
3 + -11x + 3x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
3 + -11x + 3x2 = 0

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

Divide each side by '3'.
1 + -3.666666667x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -3.666666667x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -3.666666667x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -3.666666667x + x2 = 0 + -1
-3.666666667x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-3.666666667x + x2 = -1

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 = -1 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667x + x2 = -1 + 3.361111114

Combine like terms: -1 + 3.361111114 = 2.361111114
3.361111114 + -3.666666667x + x2 = 2.361111114

Factor a perfect square on the left side:
(x + -1.833333334)(x + -1.833333334) = 2.361111114

Calculate the square root of the right side: 1.536590744

Break this problem into two subproblems by setting 
(x + -1.833333334) equal to 1.536590744 and -1.536590744.

Subproblem 1

x + -1.833333334 = 1.536590744 Simplifying x + -1.833333334 = 1.536590744 Reorder the terms: -1.833333334 + x = 1.536590744 Solving -1.833333334 + x = 1.536590744 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 = 1.536590744 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = 1.536590744 + 1.833333334 x = 1.536590744 + 1.833333334 Combine like terms: 1.536590744 + 1.833333334 = 3.369924078 x = 3.369924078 Simplifying x = 3.369924078

Subproblem 2

x + -1.833333334 = -1.536590744 Simplifying x + -1.833333334 = -1.536590744 Reorder the terms: -1.833333334 + x = -1.536590744 Solving -1.833333334 + x = -1.536590744 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 = -1.536590744 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = -1.536590744 + 1.833333334 x = -1.536590744 + 1.833333334 Combine like terms: -1.536590744 + 1.833333334 = 0.29674259 x = 0.29674259 Simplifying x = 0.29674259

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.369924078, 0.29674259}

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