(6a^2+5a+10)-(4a^2+6a+12)=0

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Solution for (6a^2+5a+10)-(4a^2+6a+12)=0 equation:


Simplifying
(6a2 + 5a + 10) + -1(4a2 + 6a + 12) = 0

Reorder the terms:
(10 + 5a + 6a2) + -1(4a2 + 6a + 12) = 0

Remove parenthesis around (10 + 5a + 6a2)
10 + 5a + 6a2 + -1(4a2 + 6a + 12) = 0

Reorder the terms:
10 + 5a + 6a2 + -1(12 + 6a + 4a2) = 0
10 + 5a + 6a2 + (12 * -1 + 6a * -1 + 4a2 * -1) = 0
10 + 5a + 6a2 + (-12 + -6a + -4a2) = 0

Reorder the terms:
10 + -12 + 5a + -6a + 6a2 + -4a2 = 0

Combine like terms: 10 + -12 = -2
-2 + 5a + -6a + 6a2 + -4a2 = 0

Combine like terms: 5a + -6a = -1a
-2 + -1a + 6a2 + -4a2 = 0

Combine like terms: 6a2 + -4a2 = 2a2
-2 + -1a + 2a2 = 0

Solving
-2 + -1a + 2a2 = 0

Solving for variable 'a'.

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

Divide each side by '2'.
-1 + -0.5a + a2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.5a + 1 + a2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.5a + a2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.5a + a2 = 0 + 1
-0.5a + a2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.5a + a2 = 1

The a term is -0.5a.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5a + 0.0625 + a2 = 1 + 0.0625

Reorder the terms:
0.0625 + -0.5a + a2 = 1 + 0.0625

Combine like terms: 1 + 0.0625 = 1.0625
0.0625 + -0.5a + a2 = 1.0625

Factor a perfect square on the left side:
(a + -0.25)(a + -0.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(a + -0.25) equal to 1.030776406 and -1.030776406.

Subproblem 1

a + -0.25 = 1.030776406 Simplifying a + -0.25 = 1.030776406 Reorder the terms: -0.25 + a = 1.030776406 Solving -0.25 + a = 1.030776406 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = 1.030776406 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = 1.030776406 + 0.25 a = 1.030776406 + 0.25 Combine like terms: 1.030776406 + 0.25 = 1.280776406 a = 1.280776406 Simplifying a = 1.280776406

Subproblem 2

a + -0.25 = -1.030776406 Simplifying a + -0.25 = -1.030776406 Reorder the terms: -0.25 + a = -1.030776406 Solving -0.25 + a = -1.030776406 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = -1.030776406 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = -1.030776406 + 0.25 a = -1.030776406 + 0.25 Combine like terms: -1.030776406 + 0.25 = -0.780776406 a = -0.780776406 Simplifying a = -0.780776406

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.280776406, -0.780776406}

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