3a^2-12=-7a

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Solution for 3a^2-12=-7a equation:


Simplifying
3a2 + -12 = -7a

Reorder the terms:
-12 + 3a2 = -7a

Solving
-12 + 3a2 = -7a

Solving for variable 'a'.

Reorder the terms:
-12 + 7a + 3a2 = -7a + 7a

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

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

Divide each side by '3'.
-4 + 2.333333333a + a2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 2.333333333a + 4 + a2 = 0 + 4

Reorder the terms:
-4 + 4 + 2.333333333a + a2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 2.333333333a + a2 = 0 + 4
2.333333333a + a2 = 0 + 4

Combine like terms: 0 + 4 = 4
2.333333333a + a2 = 4

The a term is 2.333333333a.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333a + 1.361111112 + a2 = 4 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333a + a2 = 4 + 1.361111112

Combine like terms: 4 + 1.361111112 = 5.361111112
1.361111112 + 2.333333333a + a2 = 5.361111112

Factor a perfect square on the left side:
(a + 1.166666667)(a + 1.166666667) = 5.361111112

Calculate the square root of the right side: 2.315407332

Break this problem into two subproblems by setting 
(a + 1.166666667) equal to 2.315407332 and -2.315407332.

Subproblem 1

a + 1.166666667 = 2.315407332 Simplifying a + 1.166666667 = 2.315407332 Reorder the terms: 1.166666667 + a = 2.315407332 Solving 1.166666667 + a = 2.315407332 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + a = 2.315407332 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + a = 2.315407332 + -1.166666667 a = 2.315407332 + -1.166666667 Combine like terms: 2.315407332 + -1.166666667 = 1.148740665 a = 1.148740665 Simplifying a = 1.148740665

Subproblem 2

a + 1.166666667 = -2.315407332 Simplifying a + 1.166666667 = -2.315407332 Reorder the terms: 1.166666667 + a = -2.315407332 Solving 1.166666667 + a = -2.315407332 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + a = -2.315407332 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + a = -2.315407332 + -1.166666667 a = -2.315407332 + -1.166666667 Combine like terms: -2.315407332 + -1.166666667 = -3.482073999 a = -3.482073999 Simplifying a = -3.482073999

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.148740665, -3.482073999}

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