464-a^2=a-3

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Solution for 464-a^2=a-3 equation:


Simplifying
464 + -1a2 = a + -3

Reorder the terms:
464 + -1a2 = -3 + a

Solving
464 + -1a2 = -3 + a

Solving for variable 'a'.

Reorder the terms:
464 + 3 + -1a + -1a2 = -3 + a + 3 + -1a

Combine like terms: 464 + 3 = 467
467 + -1a + -1a2 = -3 + a + 3 + -1a

Reorder the terms:
467 + -1a + -1a2 = -3 + 3 + a + -1a

Combine like terms: -3 + 3 = 0
467 + -1a + -1a2 = 0 + a + -1a
467 + -1a + -1a2 = a + -1a

Combine like terms: a + -1a = 0
467 + -1a + -1a2 = 0

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

Divide each side by '-1'.
-467 + a + a2 = 0

Move the constant term to the right:

Add '467' to each side of the equation.
-467 + a + 467 + a2 = 0 + 467

Reorder the terms:
-467 + 467 + a + a2 = 0 + 467

Combine like terms: -467 + 467 = 0
0 + a + a2 = 0 + 467
a + a2 = 0 + 467

Combine like terms: 0 + 467 = 467
a + a2 = 467

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

Add '0.25' to each side of the equation.
 + 0.25 + a2 = 467 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + a2 = 467 + 0.25

Combine like terms: 467 + 0.25 = 467.25
1.25 + a2 = 467.25

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 467.25

Calculate the square root of the right side: 21.615966321

Break this problem into two subproblems by setting 
(a + 0.5) equal to 21.615966321 and -21.615966321.

Subproblem 1

a + 0.5 = 21.615966321 Simplifying a + 0.5 = 21.615966321 Reorder the terms: 0.5 + a = 21.615966321 Solving 0.5 + a = 21.615966321 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 21.615966321 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 21.615966321 + -0.5 a = 21.615966321 + -0.5 Combine like terms: 21.615966321 + -0.5 = 21.115966321 a = 21.115966321 Simplifying a = 21.115966321

Subproblem 2

a + 0.5 = -21.615966321 Simplifying a + 0.5 = -21.615966321 Reorder the terms: 0.5 + a = -21.615966321 Solving 0.5 + a = -21.615966321 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -21.615966321 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -21.615966321 + -0.5 a = -21.615966321 + -0.5 Combine like terms: -21.615966321 + -0.5 = -22.115966321 a = -22.115966321 Simplifying a = -22.115966321

Solution

The solution to the problem is based on the solutions from the subproblems. a = {21.115966321, -22.115966321}

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