0=4.9t^2+12t-69

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Solution for 0=4.9t^2+12t-69 equation:


Simplifying
0 = 4.9t2 + 12t + -69

Reorder the terms:
0 = -69 + 12t + 4.9t2

Solving
0 = -69 + 12t + 4.9t2

Solving for variable 't'.

Combine like terms: 0 + 69 = 69
69 + -12t + -4.9t2 = -69 + 12t + 4.9t2 + 69 + -12t + -4.9t2

Reorder the terms:
69 + -12t + -4.9t2 = -69 + 69 + 12t + -12t + 4.9t2 + -4.9t2

Combine like terms: -69 + 69 = 0
69 + -12t + -4.9t2 = 0 + 12t + -12t + 4.9t2 + -4.9t2
69 + -12t + -4.9t2 = 12t + -12t + 4.9t2 + -4.9t2

Combine like terms: 12t + -12t = 0
69 + -12t + -4.9t2 = 0 + 4.9t2 + -4.9t2
69 + -12t + -4.9t2 = 4.9t2 + -4.9t2

Combine like terms: 4.9t2 + -4.9t2 = 0.0
69 + -12t + -4.9t2 = 0.0

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

Divide each side by '-4.9'.
-14.08163265 + 2.448979592t + t2 = 0

Move the constant term to the right:

Add '14.08163265' to each side of the equation.
-14.08163265 + 2.448979592t + 14.08163265 + t2 = 0 + 14.08163265

Reorder the terms:
-14.08163265 + 14.08163265 + 2.448979592t + t2 = 0 + 14.08163265

Combine like terms: -14.08163265 + 14.08163265 = 0.00000000
0.00000000 + 2.448979592t + t2 = 0 + 14.08163265
2.448979592t + t2 = 0 + 14.08163265

Combine like terms: 0 + 14.08163265 = 14.08163265
2.448979592t + t2 = 14.08163265

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 = 14.08163265 + 1.499375261

Reorder the terms:
1.499375261 + 2.448979592t + t2 = 14.08163265 + 1.499375261

Combine like terms: 14.08163265 + 1.499375261 = 15.581007911
1.499375261 + 2.448979592t + t2 = 15.581007911

Factor a perfect square on the left side:
(t + 1.224489796)(t + 1.224489796) = 15.581007911

Calculate the square root of the right side: 3.947278545

Break this problem into two subproblems by setting 
(t + 1.224489796) equal to 3.947278545 and -3.947278545.

Subproblem 1

t + 1.224489796 = 3.947278545 Simplifying t + 1.224489796 = 3.947278545 Reorder the terms: 1.224489796 + t = 3.947278545 Solving 1.224489796 + t = 3.947278545 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 = 3.947278545 + -1.224489796 Combine like terms: 1.224489796 + -1.224489796 = 0.000000000 0.000000000 + t = 3.947278545 + -1.224489796 t = 3.947278545 + -1.224489796 Combine like terms: 3.947278545 + -1.224489796 = 2.722788749 t = 2.722788749 Simplifying t = 2.722788749

Subproblem 2

t + 1.224489796 = -3.947278545 Simplifying t + 1.224489796 = -3.947278545 Reorder the terms: 1.224489796 + t = -3.947278545 Solving 1.224489796 + t = -3.947278545 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 = -3.947278545 + -1.224489796 Combine like terms: 1.224489796 + -1.224489796 = 0.000000000 0.000000000 + t = -3.947278545 + -1.224489796 t = -3.947278545 + -1.224489796 Combine like terms: -3.947278545 + -1.224489796 = -5.171768341 t = -5.171768341 Simplifying t = -5.171768341

Solution

The solution to the problem is based on the solutions from the subproblems. t = {2.722788749, -5.171768341}

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