-5312.3+523.1a-7.5a^2=0

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Solution for -5312.3+523.1a-7.5a^2=0 equation:


Simplifying
-5312.3 + 523.1a + -7.5a2 = 0

Solving
-5312.3 + 523.1a + -7.5a2 = 0

Solving for variable 'a'.

Begin completing the square.  Divide all terms by
-7.5 the coefficient of the squared term: 

Divide each side by '-7.5'.
708.3066667 + -69.74666667a + a2 = 0

Move the constant term to the right:

Add '-708.3066667' to each side of the equation.
708.3066667 + -69.74666667a + -708.3066667 + a2 = 0 + -708.3066667

Reorder the terms:
708.3066667 + -708.3066667 + -69.74666667a + a2 = 0 + -708.3066667

Combine like terms: 708.3066667 + -708.3066667 = 0.0000000
0.0000000 + -69.74666667a + a2 = 0 + -708.3066667
-69.74666667a + a2 = 0 + -708.3066667

Combine like terms: 0 + -708.3066667 = -708.3066667
-69.74666667a + a2 = -708.3066667

The a term is -69.74666667a.  Take half its coefficient (-34.87333334).
Square it (1216.149378) and add it to both sides.

Add '1216.149378' to each side of the equation.
-69.74666667a + 1216.149378 + a2 = -708.3066667 + 1216.149378

Reorder the terms:
1216.149378 + -69.74666667a + a2 = -708.3066667 + 1216.149378

Combine like terms: -708.3066667 + 1216.149378 = 507.8427113
1216.149378 + -69.74666667a + a2 = 507.8427113

Factor a perfect square on the left side:
(a + -34.87333334)(a + -34.87333334) = 507.8427113

Calculate the square root of the right side: 22.53536579

Break this problem into two subproblems by setting 
(a + -34.87333334) equal to 22.53536579 and -22.53536579.

Subproblem 1

a + -34.87333334 = 22.53536579 Simplifying a + -34.87333334 = 22.53536579 Reorder the terms: -34.87333334 + a = 22.53536579 Solving -34.87333334 + a = 22.53536579 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '34.87333334' to each side of the equation. -34.87333334 + 34.87333334 + a = 22.53536579 + 34.87333334 Combine like terms: -34.87333334 + 34.87333334 = 0.00000000 0.00000000 + a = 22.53536579 + 34.87333334 a = 22.53536579 + 34.87333334 Combine like terms: 22.53536579 + 34.87333334 = 57.40869913 a = 57.40869913 Simplifying a = 57.40869913

Subproblem 2

a + -34.87333334 = -22.53536579 Simplifying a + -34.87333334 = -22.53536579 Reorder the terms: -34.87333334 + a = -22.53536579 Solving -34.87333334 + a = -22.53536579 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '34.87333334' to each side of the equation. -34.87333334 + 34.87333334 + a = -22.53536579 + 34.87333334 Combine like terms: -34.87333334 + 34.87333334 = 0.00000000 0.00000000 + a = -22.53536579 + 34.87333334 a = -22.53536579 + 34.87333334 Combine like terms: -22.53536579 + 34.87333334 = 12.33796755 a = 12.33796755 Simplifying a = 12.33796755

Solution

The solution to the problem is based on the solutions from the subproblems. a = {57.40869913, 12.33796755}

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