3a-a^2=-15

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Solution for 3a-a^2=-15 equation:


Simplifying
3a + -1a2 = -15

Solving
3a + -1a2 = -15

Solving for variable 'a'.

Reorder the terms:
15 + 3a + -1a2 = -15 + 15

Combine like terms: -15 + 15 = 0
15 + 3a + -1a2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-15 + -3a + a2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + -3a + 15 + a2 = 0 + 15

Reorder the terms:
-15 + 15 + -3a + a2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + -3a + a2 = 0 + 15
-3a + a2 = 0 + 15

Combine like terms: 0 + 15 = 15
-3a + a2 = 15

The a term is -3a.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3a + 2.25 + a2 = 15 + 2.25

Reorder the terms:
2.25 + -3a + a2 = 15 + 2.25

Combine like terms: 15 + 2.25 = 17.25
2.25 + -3a + a2 = 17.25

Factor a perfect square on the left side:
(a + -1.5)(a + -1.5) = 17.25

Calculate the square root of the right side: 4.153311931

Break this problem into two subproblems by setting 
(a + -1.5) equal to 4.153311931 and -4.153311931.

Subproblem 1

a + -1.5 = 4.153311931 Simplifying a + -1.5 = 4.153311931 Reorder the terms: -1.5 + a = 4.153311931 Solving -1.5 + a = 4.153311931 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + a = 4.153311931 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = 4.153311931 + 1.5 a = 4.153311931 + 1.5 Combine like terms: 4.153311931 + 1.5 = 5.653311931 a = 5.653311931 Simplifying a = 5.653311931

Subproblem 2

a + -1.5 = -4.153311931 Simplifying a + -1.5 = -4.153311931 Reorder the terms: -1.5 + a = -4.153311931 Solving -1.5 + a = -4.153311931 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + a = -4.153311931 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = -4.153311931 + 1.5 a = -4.153311931 + 1.5 Combine like terms: -4.153311931 + 1.5 = -2.653311931 a = -2.653311931 Simplifying a = -2.653311931

Solution

The solution to the problem is based on the solutions from the subproblems. a = {5.653311931, -2.653311931}

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