(2c+7)(3c-4)=12

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Solution for (2c+7)(3c-4)=12 equation:


Simplifying
(2c + 7)(3c + -4) = 12

Reorder the terms:
(7 + 2c)(3c + -4) = 12

Reorder the terms:
(7 + 2c)(-4 + 3c) = 12

Multiply (7 + 2c) * (-4 + 3c)
(7(-4 + 3c) + 2c * (-4 + 3c)) = 12
((-4 * 7 + 3c * 7) + 2c * (-4 + 3c)) = 12
((-28 + 21c) + 2c * (-4 + 3c)) = 12
(-28 + 21c + (-4 * 2c + 3c * 2c)) = 12
(-28 + 21c + (-8c + 6c2)) = 12

Combine like terms: 21c + -8c = 13c
(-28 + 13c + 6c2) = 12

Solving
-28 + 13c + 6c2 = 12

Solving for variable 'c'.

Reorder the terms:
-28 + -12 + 13c + 6c2 = 12 + -12

Combine like terms: -28 + -12 = -40
-40 + 13c + 6c2 = 12 + -12

Combine like terms: 12 + -12 = 0
-40 + 13c + 6c2 = 0

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

Divide each side by '6'.
-6.666666667 + 2.166666667c + c2 = 0

Move the constant term to the right:

Add '6.666666667' to each side of the equation.
-6.666666667 + 2.166666667c + 6.666666667 + c2 = 0 + 6.666666667

Reorder the terms:
-6.666666667 + 6.666666667 + 2.166666667c + c2 = 0 + 6.666666667

Combine like terms: -6.666666667 + 6.666666667 = 0.000000000
0.000000000 + 2.166666667c + c2 = 0 + 6.666666667
2.166666667c + c2 = 0 + 6.666666667

Combine like terms: 0 + 6.666666667 = 6.666666667
2.166666667c + c2 = 6.666666667

The c term is 2.166666667c.  Take half its coefficient (1.083333334).
Square it (1.173611113) and add it to both sides.

Add '1.173611113' to each side of the equation.
2.166666667c + 1.173611113 + c2 = 6.666666667 + 1.173611113

Reorder the terms:
1.173611113 + 2.166666667c + c2 = 6.666666667 + 1.173611113

Combine like terms: 6.666666667 + 1.173611113 = 7.84027778
1.173611113 + 2.166666667c + c2 = 7.84027778

Factor a perfect square on the left side:
(c + 1.083333334)(c + 1.083333334) = 7.84027778

Calculate the square root of the right side: 2.800049603

Break this problem into two subproblems by setting 
(c + 1.083333334) equal to 2.800049603 and -2.800049603.

Subproblem 1

c + 1.083333334 = 2.800049603 Simplifying c + 1.083333334 = 2.800049603 Reorder the terms: 1.083333334 + c = 2.800049603 Solving 1.083333334 + c = 2.800049603 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + c = 2.800049603 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + c = 2.800049603 + -1.083333334 c = 2.800049603 + -1.083333334 Combine like terms: 2.800049603 + -1.083333334 = 1.716716269 c = 1.716716269 Simplifying c = 1.716716269

Subproblem 2

c + 1.083333334 = -2.800049603 Simplifying c + 1.083333334 = -2.800049603 Reorder the terms: 1.083333334 + c = -2.800049603 Solving 1.083333334 + c = -2.800049603 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + c = -2.800049603 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + c = -2.800049603 + -1.083333334 c = -2.800049603 + -1.083333334 Combine like terms: -2.800049603 + -1.083333334 = -3.883382937 c = -3.883382937 Simplifying c = -3.883382937

Solution

The solution to the problem is based on the solutions from the subproblems. c = {1.716716269, -3.883382937}

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