(3x+2)(4x+3)=122

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Solution for (3x+2)(4x+3)=122 equation:


Simplifying
(3x + 2)(4x + 3) = 122

Reorder the terms:
(2 + 3x)(4x + 3) = 122

Reorder the terms:
(2 + 3x)(3 + 4x) = 122

Multiply (2 + 3x) * (3 + 4x)
(2(3 + 4x) + 3x * (3 + 4x)) = 122
((3 * 2 + 4x * 2) + 3x * (3 + 4x)) = 122
((6 + 8x) + 3x * (3 + 4x)) = 122
(6 + 8x + (3 * 3x + 4x * 3x)) = 122
(6 + 8x + (9x + 12x2)) = 122

Combine like terms: 8x + 9x = 17x
(6 + 17x + 12x2) = 122

Solving
6 + 17x + 12x2 = 122

Solving for variable 'x'.

Reorder the terms:
6 + -122 + 17x + 12x2 = 122 + -122

Combine like terms: 6 + -122 = -116
-116 + 17x + 12x2 = 122 + -122

Combine like terms: 122 + -122 = 0
-116 + 17x + 12x2 = 0

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

Divide each side by '12'.
-9.666666667 + 1.416666667x + x2 = 0

Move the constant term to the right:

Add '9.666666667' to each side of the equation.
-9.666666667 + 1.416666667x + 9.666666667 + x2 = 0 + 9.666666667

Reorder the terms:
-9.666666667 + 9.666666667 + 1.416666667x + x2 = 0 + 9.666666667

Combine like terms: -9.666666667 + 9.666666667 = 0.000000000
0.000000000 + 1.416666667x + x2 = 0 + 9.666666667
1.416666667x + x2 = 0 + 9.666666667

Combine like terms: 0 + 9.666666667 = 9.666666667
1.416666667x + x2 = 9.666666667

The x term is 1.416666667x.  Take half its coefficient (0.7083333335).
Square it (0.5017361113) and add it to both sides.

Add '0.5017361113' to each side of the equation.
1.416666667x + 0.5017361113 + x2 = 9.666666667 + 0.5017361113

Reorder the terms:
0.5017361113 + 1.416666667x + x2 = 9.666666667 + 0.5017361113

Combine like terms: 9.666666667 + 0.5017361113 = 10.1684027783
0.5017361113 + 1.416666667x + x2 = 10.1684027783

Factor a perfect square on the left side:
(x + 0.7083333335)(x + 0.7083333335) = 10.1684027783

Calculate the square root of the right side: 3.188793311

Break this problem into two subproblems by setting 
(x + 0.7083333335) equal to 3.188793311 and -3.188793311.

Subproblem 1

x + 0.7083333335 = 3.188793311 Simplifying x + 0.7083333335 = 3.188793311 Reorder the terms: 0.7083333335 + x = 3.188793311 Solving 0.7083333335 + x = 3.188793311 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7083333335' to each side of the equation. 0.7083333335 + -0.7083333335 + x = 3.188793311 + -0.7083333335 Combine like terms: 0.7083333335 + -0.7083333335 = 0.0000000000 0.0000000000 + x = 3.188793311 + -0.7083333335 x = 3.188793311 + -0.7083333335 Combine like terms: 3.188793311 + -0.7083333335 = 2.4804599775 x = 2.4804599775 Simplifying x = 2.4804599775

Subproblem 2

x + 0.7083333335 = -3.188793311 Simplifying x + 0.7083333335 = -3.188793311 Reorder the terms: 0.7083333335 + x = -3.188793311 Solving 0.7083333335 + x = -3.188793311 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7083333335' to each side of the equation. 0.7083333335 + -0.7083333335 + x = -3.188793311 + -0.7083333335 Combine like terms: 0.7083333335 + -0.7083333335 = 0.0000000000 0.0000000000 + x = -3.188793311 + -0.7083333335 x = -3.188793311 + -0.7083333335 Combine like terms: -3.188793311 + -0.7083333335 = -3.8971266445 x = -3.8971266445 Simplifying x = -3.8971266445

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.4804599775, -3.8971266445}

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