(9a-8)(5a+6)=180

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Solution for (9a-8)(5a+6)=180 equation:


Simplifying
(9a + -8)(5a + 6) = 180

Reorder the terms:
(-8 + 9a)(5a + 6) = 180

Reorder the terms:
(-8 + 9a)(6 + 5a) = 180

Multiply (-8 + 9a) * (6 + 5a)
(-8(6 + 5a) + 9a * (6 + 5a)) = 180
((6 * -8 + 5a * -8) + 9a * (6 + 5a)) = 180
((-48 + -40a) + 9a * (6 + 5a)) = 180
(-48 + -40a + (6 * 9a + 5a * 9a)) = 180
(-48 + -40a + (54a + 45a2)) = 180

Combine like terms: -40a + 54a = 14a
(-48 + 14a + 45a2) = 180

Solving
-48 + 14a + 45a2 = 180

Solving for variable 'a'.

Reorder the terms:
-48 + -180 + 14a + 45a2 = 180 + -180

Combine like terms: -48 + -180 = -228
-228 + 14a + 45a2 = 180 + -180

Combine like terms: 180 + -180 = 0
-228 + 14a + 45a2 = 0

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

Divide each side by '45'.
-5.066666667 + 0.3111111111a + a2 = 0

Move the constant term to the right:

Add '5.066666667' to each side of the equation.
-5.066666667 + 0.3111111111a + 5.066666667 + a2 = 0 + 5.066666667

Reorder the terms:
-5.066666667 + 5.066666667 + 0.3111111111a + a2 = 0 + 5.066666667

Combine like terms: -5.066666667 + 5.066666667 = 0.000000000
0.000000000 + 0.3111111111a + a2 = 0 + 5.066666667
0.3111111111a + a2 = 0 + 5.066666667

Combine like terms: 0 + 5.066666667 = 5.066666667
0.3111111111a + a2 = 5.066666667

The a term is 0.3111111111a.  Take half its coefficient (0.1555555556).
Square it (0.02419753088) and add it to both sides.

Add '0.02419753088' to each side of the equation.
0.3111111111a + 0.02419753088 + a2 = 5.066666667 + 0.02419753088

Reorder the terms:
0.02419753088 + 0.3111111111a + a2 = 5.066666667 + 0.02419753088

Combine like terms: 5.066666667 + 0.02419753088 = 5.09086419788
0.02419753088 + 0.3111111111a + a2 = 5.09086419788

Factor a perfect square on the left side:
(a + 0.1555555556)(a + 0.1555555556) = 5.09086419788

Calculate the square root of the right side: 2.256294351

Break this problem into two subproblems by setting 
(a + 0.1555555556) equal to 2.256294351 and -2.256294351.

Subproblem 1

a + 0.1555555556 = 2.256294351 Simplifying a + 0.1555555556 = 2.256294351 Reorder the terms: 0.1555555556 + a = 2.256294351 Solving 0.1555555556 + a = 2.256294351 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1555555556' to each side of the equation. 0.1555555556 + -0.1555555556 + a = 2.256294351 + -0.1555555556 Combine like terms: 0.1555555556 + -0.1555555556 = 0.0000000000 0.0000000000 + a = 2.256294351 + -0.1555555556 a = 2.256294351 + -0.1555555556 Combine like terms: 2.256294351 + -0.1555555556 = 2.1007387954 a = 2.1007387954 Simplifying a = 2.1007387954

Subproblem 2

a + 0.1555555556 = -2.256294351 Simplifying a + 0.1555555556 = -2.256294351 Reorder the terms: 0.1555555556 + a = -2.256294351 Solving 0.1555555556 + a = -2.256294351 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1555555556' to each side of the equation. 0.1555555556 + -0.1555555556 + a = -2.256294351 + -0.1555555556 Combine like terms: 0.1555555556 + -0.1555555556 = 0.0000000000 0.0000000000 + a = -2.256294351 + -0.1555555556 a = -2.256294351 + -0.1555555556 Combine like terms: -2.256294351 + -0.1555555556 = -2.4118499066 a = -2.4118499066 Simplifying a = -2.4118499066

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.1007387954, -2.4118499066}

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