(7a^2-9a+6)-(12a^2-4a+3)=0

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Solution for (7a^2-9a+6)-(12a^2-4a+3)=0 equation:


Simplifying
(7a2 + -9a + 6) + -1(12a2 + -4a + 3) = 0

Reorder the terms:
(6 + -9a + 7a2) + -1(12a2 + -4a + 3) = 0

Remove parenthesis around (6 + -9a + 7a2)
6 + -9a + 7a2 + -1(12a2 + -4a + 3) = 0

Reorder the terms:
6 + -9a + 7a2 + -1(3 + -4a + 12a2) = 0
6 + -9a + 7a2 + (3 * -1 + -4a * -1 + 12a2 * -1) = 0
6 + -9a + 7a2 + (-3 + 4a + -12a2) = 0

Reorder the terms:
6 + -3 + -9a + 4a + 7a2 + -12a2 = 0

Combine like terms: 6 + -3 = 3
3 + -9a + 4a + 7a2 + -12a2 = 0

Combine like terms: -9a + 4a = -5a
3 + -5a + 7a2 + -12a2 = 0

Combine like terms: 7a2 + -12a2 = -5a2
3 + -5a + -5a2 = 0

Solving
3 + -5a + -5a2 = 0

Solving for variable 'a'.

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

Divide each side by '-5'.
-0.6 + a + a2 = 0

Move the constant term to the right:

Add '0.6' to each side of the equation.
-0.6 + a + 0.6 + a2 = 0 + 0.6

Reorder the terms:
-0.6 + 0.6 + a + a2 = 0 + 0.6

Combine like terms: -0.6 + 0.6 = 0.0
0.0 + a + a2 = 0 + 0.6
a + a2 = 0 + 0.6

Combine like terms: 0 + 0.6 = 0.6
a + a2 = 0.6

The a term is a.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + a2 = 0.6 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + a2 = 0.6 + 0.25

Combine like terms: 0.6 + 0.25 = 0.85
1.25 + a2 = 0.85

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 0.85

Calculate the square root of the right side: 0.921954446

Break this problem into two subproblems by setting 
(a + 0.5) equal to 0.921954446 and -0.921954446.

Subproblem 1

a + 0.5 = 0.921954446 Simplifying a + 0.5 = 0.921954446 Reorder the terms: 0.5 + a = 0.921954446 Solving 0.5 + a = 0.921954446 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 0.921954446 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 0.921954446 + -0.5 a = 0.921954446 + -0.5 Combine like terms: 0.921954446 + -0.5 = 0.421954446 a = 0.421954446 Simplifying a = 0.421954446

Subproblem 2

a + 0.5 = -0.921954446 Simplifying a + 0.5 = -0.921954446 Reorder the terms: 0.5 + a = -0.921954446 Solving 0.5 + a = -0.921954446 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -0.921954446 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -0.921954446 + -0.5 a = -0.921954446 + -0.5 Combine like terms: -0.921954446 + -0.5 = -1.421954446 a = -1.421954446 Simplifying a = -1.421954446

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.421954446, -1.421954446}

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