115=37.3x+4.9x^2

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Solution for 115=37.3x+4.9x^2 equation:


Simplifying
115 = 37.3x + 4.9x2

Solving
115 = 37.3x + 4.9x2

Solving for variable 'x'.

Reorder the terms:
115 + -37.3x + -4.9x2 = 37.3x + -37.3x + 4.9x2 + -4.9x2

Combine like terms: 37.3x + -37.3x = 0.0
115 + -37.3x + -4.9x2 = 0.0 + 4.9x2 + -4.9x2
115 + -37.3x + -4.9x2 = 4.9x2 + -4.9x2

Combine like terms: 4.9x2 + -4.9x2 = 0.0
115 + -37.3x + -4.9x2 = 0.0

Begin completing the square.  Divide all terms by
-4.9 the coefficient of the squared term: 

Divide each side by '-4.9'.
-23.46938776 + 7.612244898x + x2 = 0

Move the constant term to the right:

Add '23.46938776' to each side of the equation.
-23.46938776 + 7.612244898x + 23.46938776 + x2 = 0 + 23.46938776

Reorder the terms:
-23.46938776 + 23.46938776 + 7.612244898x + x2 = 0 + 23.46938776

Combine like terms: -23.46938776 + 23.46938776 = 0.00000000
0.00000000 + 7.612244898x + x2 = 0 + 23.46938776
7.612244898x + x2 = 0 + 23.46938776

Combine like terms: 0 + 23.46938776 = 23.46938776
7.612244898x + x2 = 23.46938776

The x term is 7.612244898x.  Take half its coefficient (3.806122449).
Square it (14.48656810) and add it to both sides.

Add '14.48656810' to each side of the equation.
7.612244898x + 14.48656810 + x2 = 23.46938776 + 14.48656810

Reorder the terms:
14.48656810 + 7.612244898x + x2 = 23.46938776 + 14.48656810

Combine like terms: 23.46938776 + 14.48656810 = 37.95595586
14.48656810 + 7.612244898x + x2 = 37.95595586

Factor a perfect square on the left side:
(x + 3.806122449)(x + 3.806122449) = 37.95595586

Calculate the square root of the right side: 6.160840516

Break this problem into two subproblems by setting 
(x + 3.806122449) equal to 6.160840516 and -6.160840516.

Subproblem 1

x + 3.806122449 = 6.160840516 Simplifying x + 3.806122449 = 6.160840516 Reorder the terms: 3.806122449 + x = 6.160840516 Solving 3.806122449 + x = 6.160840516 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.806122449' to each side of the equation. 3.806122449 + -3.806122449 + x = 6.160840516 + -3.806122449 Combine like terms: 3.806122449 + -3.806122449 = 0.000000000 0.000000000 + x = 6.160840516 + -3.806122449 x = 6.160840516 + -3.806122449 Combine like terms: 6.160840516 + -3.806122449 = 2.354718067 x = 2.354718067 Simplifying x = 2.354718067

Subproblem 2

x + 3.806122449 = -6.160840516 Simplifying x + 3.806122449 = -6.160840516 Reorder the terms: 3.806122449 + x = -6.160840516 Solving 3.806122449 + x = -6.160840516 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.806122449' to each side of the equation. 3.806122449 + -3.806122449 + x = -6.160840516 + -3.806122449 Combine like terms: 3.806122449 + -3.806122449 = 0.000000000 0.000000000 + x = -6.160840516 + -3.806122449 x = -6.160840516 + -3.806122449 Combine like terms: -6.160840516 + -3.806122449 = -9.966962965 x = -9.966962965 Simplifying x = -9.966962965

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.354718067, -9.966962965}

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