(3x-4)(2x+3)=78

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Solution for (3x-4)(2x+3)=78 equation:


Simplifying
(3x + -4)(2x + 3) = 78

Reorder the terms:
(-4 + 3x)(2x + 3) = 78

Reorder the terms:
(-4 + 3x)(3 + 2x) = 78

Multiply (-4 + 3x) * (3 + 2x)
(-4(3 + 2x) + 3x * (3 + 2x)) = 78
((3 * -4 + 2x * -4) + 3x * (3 + 2x)) = 78
((-12 + -8x) + 3x * (3 + 2x)) = 78
(-12 + -8x + (3 * 3x + 2x * 3x)) = 78
(-12 + -8x + (9x + 6x2)) = 78

Combine like terms: -8x + 9x = 1x
(-12 + 1x + 6x2) = 78

Solving
-12 + 1x + 6x2 = 78

Solving for variable 'x'.

Reorder the terms:
-12 + -78 + 1x + 6x2 = 78 + -78

Combine like terms: -12 + -78 = -90
-90 + 1x + 6x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-90 + 1x + 6x2 = 0

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

Divide each side by '6'.
-15 + 0.1666666667x + x2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 0.1666666667x + 15 + x2 = 0 + 15

Reorder the terms:
-15 + 15 + 0.1666666667x + x2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 0.1666666667x + x2 = 0 + 15
0.1666666667x + x2 = 0 + 15

Combine like terms: 0 + 15 = 15
0.1666666667x + x2 = 15

The x term is 0.1666666667x.  Take half its coefficient (0.08333333335).
Square it (0.006944444447) and add it to both sides.

Add '0.006944444447' to each side of the equation.
0.1666666667x + 0.006944444447 + x2 = 15 + 0.006944444447

Reorder the terms:
0.006944444447 + 0.1666666667x + x2 = 15 + 0.006944444447

Combine like terms: 15 + 0.006944444447 = 15.006944444447
0.006944444447 + 0.1666666667x + x2 = 15.006944444447

Factor a perfect square on the left side:
(x + 0.08333333335)(x + 0.08333333335) = 15.006944444447

Calculate the square root of the right side: 3.873879766

Break this problem into two subproblems by setting 
(x + 0.08333333335) equal to 3.873879766 and -3.873879766.

Subproblem 1

x + 0.08333333335 = 3.873879766 Simplifying x + 0.08333333335 = 3.873879766 Reorder the terms: 0.08333333335 + x = 3.873879766 Solving 0.08333333335 + x = 3.873879766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = 3.873879766 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = 3.873879766 + -0.08333333335 x = 3.873879766 + -0.08333333335 Combine like terms: 3.873879766 + -0.08333333335 = 3.79054643265 x = 3.79054643265 Simplifying x = 3.79054643265

Subproblem 2

x + 0.08333333335 = -3.873879766 Simplifying x + 0.08333333335 = -3.873879766 Reorder the terms: 0.08333333335 + x = -3.873879766 Solving 0.08333333335 + x = -3.873879766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = -3.873879766 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = -3.873879766 + -0.08333333335 x = -3.873879766 + -0.08333333335 Combine like terms: -3.873879766 + -0.08333333335 = -3.95721309935 x = -3.95721309935 Simplifying x = -3.95721309935

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.79054643265, -3.95721309935}

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