(3a^2+2a-2)-(a^2-3a+7)=0

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Solution for (3a^2+2a-2)-(a^2-3a+7)=0 equation:


Simplifying
(3a2 + 2a + -2) + -1(a2 + -3a + 7) = 0

Reorder the terms:
(-2 + 2a + 3a2) + -1(a2 + -3a + 7) = 0

Remove parenthesis around (-2 + 2a + 3a2)
-2 + 2a + 3a2 + -1(a2 + -3a + 7) = 0

Reorder the terms:
-2 + 2a + 3a2 + -1(7 + -3a + a2) = 0
-2 + 2a + 3a2 + (7 * -1 + -3a * -1 + a2 * -1) = 0
-2 + 2a + 3a2 + (-7 + 3a + -1a2) = 0

Reorder the terms:
-2 + -7 + 2a + 3a + 3a2 + -1a2 = 0

Combine like terms: -2 + -7 = -9
-9 + 2a + 3a + 3a2 + -1a2 = 0

Combine like terms: 2a + 3a = 5a
-9 + 5a + 3a2 + -1a2 = 0

Combine like terms: 3a2 + -1a2 = 2a2
-9 + 5a + 2a2 = 0

Solving
-9 + 5a + 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'.
-4.5 + 2.5a + a2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + 2.5a + 4.5 + a2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + 2.5a + a2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + 2.5a + a2 = 0 + 4.5
2.5a + a2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
2.5a + a2 = 4.5

The a term is 2.5a.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5a + 1.5625 + a2 = 4.5 + 1.5625

Reorder the terms:
1.5625 + 2.5a + a2 = 4.5 + 1.5625

Combine like terms: 4.5 + 1.5625 = 6.0625
1.5625 + 2.5a + a2 = 6.0625

Factor a perfect square on the left side:
(a + 1.25)(a + 1.25) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(a + 1.25) equal to 2.46221445 and -2.46221445.

Subproblem 1

a + 1.25 = 2.46221445 Simplifying a + 1.25 = 2.46221445 Reorder the terms: 1.25 + a = 2.46221445 Solving 1.25 + a = 2.46221445 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + a = 2.46221445 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + a = 2.46221445 + -1.25 a = 2.46221445 + -1.25 Combine like terms: 2.46221445 + -1.25 = 1.21221445 a = 1.21221445 Simplifying a = 1.21221445

Subproblem 2

a + 1.25 = -2.46221445 Simplifying a + 1.25 = -2.46221445 Reorder the terms: 1.25 + a = -2.46221445 Solving 1.25 + a = -2.46221445 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + a = -2.46221445 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + a = -2.46221445 + -1.25 a = -2.46221445 + -1.25 Combine like terms: -2.46221445 + -1.25 = -3.71221445 a = -3.71221445 Simplifying a = -3.71221445

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.21221445, -3.71221445}

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