(2x+1)*(3x+2)=24

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Solution for (2x+1)*(3x+2)=24 equation:


Simplifying
(2x + 1)(3x + 2) = 24

Reorder the terms:
(1 + 2x)(3x + 2) = 24

Reorder the terms:
(1 + 2x)(2 + 3x) = 24

Multiply (1 + 2x) * (2 + 3x)
(1(2 + 3x) + 2x * (2 + 3x)) = 24
((2 * 1 + 3x * 1) + 2x * (2 + 3x)) = 24
((2 + 3x) + 2x * (2 + 3x)) = 24
(2 + 3x + (2 * 2x + 3x * 2x)) = 24
(2 + 3x + (4x + 6x2)) = 24

Combine like terms: 3x + 4x = 7x
(2 + 7x + 6x2) = 24

Solving
2 + 7x + 6x2 = 24

Solving for variable 'x'.

Reorder the terms:
2 + -24 + 7x + 6x2 = 24 + -24

Combine like terms: 2 + -24 = -22
-22 + 7x + 6x2 = 24 + -24

Combine like terms: 24 + -24 = 0
-22 + 7x + 6x2 = 0

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

Divide each side by '6'.
-3.666666667 + 1.166666667x + x2 = 0

Move the constant term to the right:

Add '3.666666667' to each side of the equation.
-3.666666667 + 1.166666667x + 3.666666667 + x2 = 0 + 3.666666667

Reorder the terms:
-3.666666667 + 3.666666667 + 1.166666667x + x2 = 0 + 3.666666667

Combine like terms: -3.666666667 + 3.666666667 = 0.000000000
0.000000000 + 1.166666667x + x2 = 0 + 3.666666667
1.166666667x + x2 = 0 + 3.666666667

Combine like terms: 0 + 3.666666667 = 3.666666667
1.166666667x + x2 = 3.666666667

The x term is 1.166666667x.  Take half its coefficient (0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
1.166666667x + 0.3402777780 + x2 = 3.666666667 + 0.3402777780

Reorder the terms:
0.3402777780 + 1.166666667x + x2 = 3.666666667 + 0.3402777780

Combine like terms: 3.666666667 + 0.3402777780 = 4.006944445
0.3402777780 + 1.166666667x + x2 = 4.006944445

Factor a perfect square on the left side:
(x + 0.5833333335)(x + 0.5833333335) = 4.006944445

Calculate the square root of the right side: 2.001735358

Break this problem into two subproblems by setting 
(x + 0.5833333335) equal to 2.001735358 and -2.001735358.

Subproblem 1

x + 0.5833333335 = 2.001735358 Simplifying x + 0.5833333335 = 2.001735358 Reorder the terms: 0.5833333335 + x = 2.001735358 Solving 0.5833333335 + x = 2.001735358 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + x = 2.001735358 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + x = 2.001735358 + -0.5833333335 x = 2.001735358 + -0.5833333335 Combine like terms: 2.001735358 + -0.5833333335 = 1.4184020245 x = 1.4184020245 Simplifying x = 1.4184020245

Subproblem 2

x + 0.5833333335 = -2.001735358 Simplifying x + 0.5833333335 = -2.001735358 Reorder the terms: 0.5833333335 + x = -2.001735358 Solving 0.5833333335 + x = -2.001735358 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + x = -2.001735358 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + x = -2.001735358 + -0.5833333335 x = -2.001735358 + -0.5833333335 Combine like terms: -2.001735358 + -0.5833333335 = -2.5850686915 x = -2.5850686915 Simplifying x = -2.5850686915

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.4184020245, -2.5850686915}

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