7a^2-4a+32a=9+12a^2

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Solution for 7a^2-4a+32a=9+12a^2 equation:


Simplifying
7a2 + -4a + 32a = 9 + 12a2

Reorder the terms:
-4a + 32a + 7a2 = 9 + 12a2

Combine like terms: -4a + 32a = 28a
28a + 7a2 = 9 + 12a2

Solving
28a + 7a2 = 9 + 12a2

Solving for variable 'a'.

Reorder the terms:
-9 + 28a + 7a2 + -12a2 = 9 + 12a2 + -9 + -12a2

Combine like terms: 7a2 + -12a2 = -5a2
-9 + 28a + -5a2 = 9 + 12a2 + -9 + -12a2

Reorder the terms:
-9 + 28a + -5a2 = 9 + -9 + 12a2 + -12a2

Combine like terms: 9 + -9 = 0
-9 + 28a + -5a2 = 0 + 12a2 + -12a2
-9 + 28a + -5a2 = 12a2 + -12a2

Combine like terms: 12a2 + -12a2 = 0
-9 + 28a + -5a2 = 0

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
1.8 + -5.6a + a2 = 0

Move the constant term to the right:

Add '-1.8' to each side of the equation.
1.8 + -5.6a + -1.8 + a2 = 0 + -1.8

Reorder the terms:
1.8 + -1.8 + -5.6a + a2 = 0 + -1.8

Combine like terms: 1.8 + -1.8 = 0.0
0.0 + -5.6a + a2 = 0 + -1.8
-5.6a + a2 = 0 + -1.8

Combine like terms: 0 + -1.8 = -1.8
-5.6a + a2 = -1.8

The a term is -5.6a.  Take half its coefficient (-2.8).
Square it (7.84) and add it to both sides.

Add '7.84' to each side of the equation.
-5.6a + 7.84 + a2 = -1.8 + 7.84

Reorder the terms:
7.84 + -5.6a + a2 = -1.8 + 7.84

Combine like terms: -1.8 + 7.84 = 6.04
7.84 + -5.6a + a2 = 6.04

Factor a perfect square on the left side:
(a + -2.8)(a + -2.8) = 6.04

Calculate the square root of the right side: 2.457641145

Break this problem into two subproblems by setting 
(a + -2.8) equal to 2.457641145 and -2.457641145.

Subproblem 1

a + -2.8 = 2.457641145 Simplifying a + -2.8 = 2.457641145 Reorder the terms: -2.8 + a = 2.457641145 Solving -2.8 + a = 2.457641145 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.8' to each side of the equation. -2.8 + 2.8 + a = 2.457641145 + 2.8 Combine like terms: -2.8 + 2.8 = 0.0 0.0 + a = 2.457641145 + 2.8 a = 2.457641145 + 2.8 Combine like terms: 2.457641145 + 2.8 = 5.257641145 a = 5.257641145 Simplifying a = 5.257641145

Subproblem 2

a + -2.8 = -2.457641145 Simplifying a + -2.8 = -2.457641145 Reorder the terms: -2.8 + a = -2.457641145 Solving -2.8 + a = -2.457641145 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.8' to each side of the equation. -2.8 + 2.8 + a = -2.457641145 + 2.8 Combine like terms: -2.8 + 2.8 = 0.0 0.0 + a = -2.457641145 + 2.8 a = -2.457641145 + 2.8 Combine like terms: -2.457641145 + 2.8 = 0.342358855 a = 0.342358855 Simplifying a = 0.342358855

Solution

The solution to the problem is based on the solutions from the subproblems. a = {5.257641145, 0.342358855}

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