(a+6)(3a-7)=04

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Solution for (a+6)(3a-7)=04 equation:


Simplifying
(a + 6)(3a + -7) = 4

Reorder the terms:
(6 + a)(3a + -7) = 4

Reorder the terms:
(6 + a)(-7 + 3a) = 4

Multiply (6 + a) * (-7 + 3a)
(6(-7 + 3a) + a(-7 + 3a)) = 4
((-7 * 6 + 3a * 6) + a(-7 + 3a)) = 4
((-42 + 18a) + a(-7 + 3a)) = 4
(-42 + 18a + (-7 * a + 3a * a)) = 4
(-42 + 18a + (-7a + 3a2)) = 4

Combine like terms: 18a + -7a = 11a
(-42 + 11a + 3a2) = 4

Solving
-42 + 11a + 3a2 = 4

Solving for variable 'a'.

Reorder the terms:
-42 + -4 + 11a + 3a2 = 4 + -4

Combine like terms: -42 + -4 = -46
-46 + 11a + 3a2 = 4 + -4

Combine like terms: 4 + -4 = 0
-46 + 11a + 3a2 = 0

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

Divide each side by '3'.
-15.33333333 + 3.666666667a + a2 = 0

Move the constant term to the right:

Add '15.33333333' to each side of the equation.
-15.33333333 + 3.666666667a + 15.33333333 + a2 = 0 + 15.33333333

Reorder the terms:
-15.33333333 + 15.33333333 + 3.666666667a + a2 = 0 + 15.33333333

Combine like terms: -15.33333333 + 15.33333333 = 0.00000000
0.00000000 + 3.666666667a + a2 = 0 + 15.33333333
3.666666667a + a2 = 0 + 15.33333333

Combine like terms: 0 + 15.33333333 = 15.33333333
3.666666667a + a2 = 15.33333333

The a term is 3.666666667a.  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.
3.666666667a + 3.361111114 + a2 = 15.33333333 + 3.361111114

Reorder the terms:
3.361111114 + 3.666666667a + a2 = 15.33333333 + 3.361111114

Combine like terms: 15.33333333 + 3.361111114 = 18.694444444
3.361111114 + 3.666666667a + a2 = 18.694444444

Factor a perfect square on the left side:
(a + 1.833333334)(a + 1.833333334) = 18.694444444

Calculate the square root of the right side: 4.323707257

Break this problem into two subproblems by setting 
(a + 1.833333334) equal to 4.323707257 and -4.323707257.

Subproblem 1

a + 1.833333334 = 4.323707257 Simplifying a + 1.833333334 = 4.323707257 Reorder the terms: 1.833333334 + a = 4.323707257 Solving 1.833333334 + a = 4.323707257 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + a = 4.323707257 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + a = 4.323707257 + -1.833333334 a = 4.323707257 + -1.833333334 Combine like terms: 4.323707257 + -1.833333334 = 2.490373923 a = 2.490373923 Simplifying a = 2.490373923

Subproblem 2

a + 1.833333334 = -4.323707257 Simplifying a + 1.833333334 = -4.323707257 Reorder the terms: 1.833333334 + a = -4.323707257 Solving 1.833333334 + a = -4.323707257 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + a = -4.323707257 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + a = -4.323707257 + -1.833333334 a = -4.323707257 + -1.833333334 Combine like terms: -4.323707257 + -1.833333334 = -6.157040591 a = -6.157040591 Simplifying a = -6.157040591

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.490373923, -6.157040591}

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