(7+3a)(5a+2)=117

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Solution for (7+3a)(5a+2)=117 equation:


Simplifying
(7 + 3a)(5a + 2) = 117

Reorder the terms:
(7 + 3a)(2 + 5a) = 117

Multiply (7 + 3a) * (2 + 5a)
(7(2 + 5a) + 3a * (2 + 5a)) = 117
((2 * 7 + 5a * 7) + 3a * (2 + 5a)) = 117
((14 + 35a) + 3a * (2 + 5a)) = 117
(14 + 35a + (2 * 3a + 5a * 3a)) = 117
(14 + 35a + (6a + 15a2)) = 117

Combine like terms: 35a + 6a = 41a
(14 + 41a + 15a2) = 117

Solving
14 + 41a + 15a2 = 117

Solving for variable 'a'.

Reorder the terms:
14 + -117 + 41a + 15a2 = 117 + -117

Combine like terms: 14 + -117 = -103
-103 + 41a + 15a2 = 117 + -117

Combine like terms: 117 + -117 = 0
-103 + 41a + 15a2 = 0

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

Divide each side by '15'.
-6.866666667 + 2.733333333a + a2 = 0

Move the constant term to the right:

Add '6.866666667' to each side of the equation.
-6.866666667 + 2.733333333a + 6.866666667 + a2 = 0 + 6.866666667

Reorder the terms:
-6.866666667 + 6.866666667 + 2.733333333a + a2 = 0 + 6.866666667

Combine like terms: -6.866666667 + 6.866666667 = 0.000000000
0.000000000 + 2.733333333a + a2 = 0 + 6.866666667
2.733333333a + a2 = 0 + 6.866666667

Combine like terms: 0 + 6.866666667 = 6.866666667
2.733333333a + a2 = 6.866666667

The a term is 2.733333333a.  Take half its coefficient (1.366666667).
Square it (1.867777779) and add it to both sides.

Add '1.867777779' to each side of the equation.
2.733333333a + 1.867777779 + a2 = 6.866666667 + 1.867777779

Reorder the terms:
1.867777779 + 2.733333333a + a2 = 6.866666667 + 1.867777779

Combine like terms: 6.866666667 + 1.867777779 = 8.734444446
1.867777779 + 2.733333333a + a2 = 8.734444446

Factor a perfect square on the left side:
(a + 1.366666667)(a + 1.366666667) = 8.734444446

Calculate the square root of the right side: 2.955409353

Break this problem into two subproblems by setting 
(a + 1.366666667) equal to 2.955409353 and -2.955409353.

Subproblem 1

a + 1.366666667 = 2.955409353 Simplifying a + 1.366666667 = 2.955409353 Reorder the terms: 1.366666667 + a = 2.955409353 Solving 1.366666667 + a = 2.955409353 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.366666667' to each side of the equation. 1.366666667 + -1.366666667 + a = 2.955409353 + -1.366666667 Combine like terms: 1.366666667 + -1.366666667 = 0.000000000 0.000000000 + a = 2.955409353 + -1.366666667 a = 2.955409353 + -1.366666667 Combine like terms: 2.955409353 + -1.366666667 = 1.588742686 a = 1.588742686 Simplifying a = 1.588742686

Subproblem 2

a + 1.366666667 = -2.955409353 Simplifying a + 1.366666667 = -2.955409353 Reorder the terms: 1.366666667 + a = -2.955409353 Solving 1.366666667 + a = -2.955409353 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.366666667' to each side of the equation. 1.366666667 + -1.366666667 + a = -2.955409353 + -1.366666667 Combine like terms: 1.366666667 + -1.366666667 = 0.000000000 0.000000000 + a = -2.955409353 + -1.366666667 a = -2.955409353 + -1.366666667 Combine like terms: -2.955409353 + -1.366666667 = -4.32207602 a = -4.32207602 Simplifying a = -4.32207602

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.588742686, -4.32207602}

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