500+12t-4.9t^2=0

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Solution for 500+12t-4.9t^2=0 equation:


Simplifying
500 + 12t + -4.9t2 = 0

Solving
500 + 12t + -4.9t2 = 0

Solving for variable 't'.

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

Divide each side by '-4.9'.
-102.0408163 + -2.448979592t + t2 = 0

Move the constant term to the right:

Add '102.0408163' to each side of the equation.
-102.0408163 + -2.448979592t + 102.0408163 + t2 = 0 + 102.0408163

Reorder the terms:
-102.0408163 + 102.0408163 + -2.448979592t + t2 = 0 + 102.0408163

Combine like terms: -102.0408163 + 102.0408163 = 0.0000000
0.0000000 + -2.448979592t + t2 = 0 + 102.0408163
-2.448979592t + t2 = 0 + 102.0408163

Combine like terms: 0 + 102.0408163 = 102.0408163
-2.448979592t + t2 = 102.0408163

The t term is -2.448979592t.  Take half its coefficient (-1.224489796).
Square it (1.499375261) and add it to both sides.

Add '1.499375261' to each side of the equation.
-2.448979592t + 1.499375261 + t2 = 102.0408163 + 1.499375261

Reorder the terms:
1.499375261 + -2.448979592t + t2 = 102.0408163 + 1.499375261

Combine like terms: 102.0408163 + 1.499375261 = 103.540191561
1.499375261 + -2.448979592t + t2 = 103.540191561

Factor a perfect square on the left side:
(t + -1.224489796)(t + -1.224489796) = 103.540191561

Calculate the square root of the right side: 10.17547009

Break this problem into two subproblems by setting 
(t + -1.224489796) equal to 10.17547009 and -10.17547009.

Subproblem 1

t + -1.224489796 = 10.17547009 Simplifying t + -1.224489796 = 10.17547009 Reorder the terms: -1.224489796 + t = 10.17547009 Solving -1.224489796 + t = 10.17547009 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.224489796' to each side of the equation. -1.224489796 + 1.224489796 + t = 10.17547009 + 1.224489796 Combine like terms: -1.224489796 + 1.224489796 = 0.000000000 0.000000000 + t = 10.17547009 + 1.224489796 t = 10.17547009 + 1.224489796 Combine like terms: 10.17547009 + 1.224489796 = 11.399959886 t = 11.399959886 Simplifying t = 11.399959886

Subproblem 2

t + -1.224489796 = -10.17547009 Simplifying t + -1.224489796 = -10.17547009 Reorder the terms: -1.224489796 + t = -10.17547009 Solving -1.224489796 + t = -10.17547009 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.224489796' to each side of the equation. -1.224489796 + 1.224489796 + t = -10.17547009 + 1.224489796 Combine like terms: -1.224489796 + 1.224489796 = 0.000000000 0.000000000 + t = -10.17547009 + 1.224489796 t = -10.17547009 + 1.224489796 Combine like terms: -10.17547009 + 1.224489796 = -8.950980294 t = -8.950980294 Simplifying t = -8.950980294

Solution

The solution to the problem is based on the solutions from the subproblems. t = {11.399959886, -8.950980294}

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