a^2=28-3a-a^2

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Solution for a^2=28-3a-a^2 equation:


Simplifying
a2 = 28 + -3a + -1a2

Solving
a2 = 28 + -3a + -1a2

Solving for variable 'a'.

Reorder the terms:
-28 + 3a + a2 + a2 = 28 + -3a + -1a2 + -28 + 3a + a2

Combine like terms: a2 + a2 = 2a2
-28 + 3a + 2a2 = 28 + -3a + -1a2 + -28 + 3a + a2

Reorder the terms:
-28 + 3a + 2a2 = 28 + -28 + -3a + 3a + -1a2 + a2

Combine like terms: 28 + -28 = 0
-28 + 3a + 2a2 = 0 + -3a + 3a + -1a2 + a2
-28 + 3a + 2a2 = -3a + 3a + -1a2 + a2

Combine like terms: -3a + 3a = 0
-28 + 3a + 2a2 = 0 + -1a2 + a2
-28 + 3a + 2a2 = -1a2 + a2

Combine like terms: -1a2 + a2 = 0
-28 + 3a + 2a2 = 0

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

Divide each side by '2'.
-14 + 1.5a + a2 = 0

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + 1.5a + 14 + a2 = 0 + 14

Reorder the terms:
-14 + 14 + 1.5a + a2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + 1.5a + a2 = 0 + 14
1.5a + a2 = 0 + 14

Combine like terms: 0 + 14 = 14
1.5a + a2 = 14

The a term is 1.5a.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5a + 0.5625 + a2 = 14 + 0.5625

Reorder the terms:
0.5625 + 1.5a + a2 = 14 + 0.5625

Combine like terms: 14 + 0.5625 = 14.5625
0.5625 + 1.5a + a2 = 14.5625

Factor a perfect square on the left side:
(a + 0.75)(a + 0.75) = 14.5625

Calculate the square root of the right side: 3.816084381

Break this problem into two subproblems by setting 
(a + 0.75) equal to 3.816084381 and -3.816084381.

Subproblem 1

a + 0.75 = 3.816084381 Simplifying a + 0.75 = 3.816084381 Reorder the terms: 0.75 + a = 3.816084381 Solving 0.75 + a = 3.816084381 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = 3.816084381 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 3.816084381 + -0.75 a = 3.816084381 + -0.75 Combine like terms: 3.816084381 + -0.75 = 3.066084381 a = 3.066084381 Simplifying a = 3.066084381

Subproblem 2

a + 0.75 = -3.816084381 Simplifying a + 0.75 = -3.816084381 Reorder the terms: 0.75 + a = -3.816084381 Solving 0.75 + a = -3.816084381 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = -3.816084381 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -3.816084381 + -0.75 a = -3.816084381 + -0.75 Combine like terms: -3.816084381 + -0.75 = -4.566084381 a = -4.566084381 Simplifying a = -4.566084381

Solution

The solution to the problem is based on the solutions from the subproblems. a = {3.066084381, -4.566084381}

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