4.9t^2-45t-90=0

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Solution for 4.9t^2-45t-90=0 equation: 


Simplifying
4.9t2 + -45t + -90 = 0

Reorder the terms:
-90 + -45t + 4.9t2 = 0

Solving
-90 + -45t + 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'.
-18.36734694 + -9.183673469t + t2 = 0

Move the constant term to the right:

Add '18.36734694' to each side of the equation.
-18.36734694 + -9.183673469t + 18.36734694 + t2 = 0 + 18.36734694

Reorder the terms:
-18.36734694 + 18.36734694 + -9.183673469t + t2 = 0 + 18.36734694

Combine like terms: -18.36734694 + 18.36734694 = 0.00000000
0.00000000 + -9.183673469t + t2 = 0 + 18.36734694
-9.183673469t + t2 = 0 + 18.36734694

Combine like terms: 0 + 18.36734694 = 18.36734694
-9.183673469t + t2 = 18.36734694

The t term is -9.183673469t.  Take half its coefficient (-4.591836735).
Square it (21.08496460) and add it to both sides.

Add '21.08496460' to each side of the equation.
-9.183673469t + 21.08496460 + t2 = 18.36734694 + 21.08496460

Reorder the terms:
21.08496460 + -9.183673469t + t2 = 18.36734694 + 21.08496460

Combine like terms: 18.36734694 + 21.08496460 = 39.45231154
21.08496460 + -9.183673469t + t2 = 39.45231154

Factor a perfect square on the left side:
(t + -4.591836735)(t + -4.591836735) = 39.45231154

Calculate the square root of the right side: 6.281107509

Break this problem into two subproblems by setting 
(t + -4.591836735) equal to 6.281107509 and -6.281107509.


Subproblem 1
t + -4.591836735 = 6.281107509

Simplifying
t + -4.591836735 = 6.281107509

Reorder the terms:
-4.591836735 + t = 6.281107509

Solving
-4.591836735 + t = 6.281107509

Solving for variable 't'.

Move all terms containing t to the left, all other terms to the right.

Add '4.591836735' to each side of the equation.
-4.591836735 + 4.591836735 + t = 6.281107509 + 4.591836735

Combine like terms: -4.591836735 + 4.591836735 = 0.000000000
0.000000000 + t = 6.281107509 + 4.591836735
t = 6.281107509 + 4.591836735

Combine like terms: 6.281107509 + 4.591836735 = 10.872944244
t = 10.872944244

Simplifying
t = 10.872944244


Subproblem 2
t + -4.591836735 = -6.281107509

Simplifying
t + -4.591836735 = -6.281107509

Reorder the terms:
-4.591836735 + t = -6.281107509

Solving
-4.591836735 + t = -6.281107509

Solving for variable 't'.

Move all terms containing t to the left, all other terms to the right.

Add '4.591836735' to each side of the equation.
-4.591836735 + 4.591836735 + t = -6.281107509 + 4.591836735

Combine like terms: -4.591836735 + 4.591836735 = 0.000000000
0.000000000 + t = -6.281107509 + 4.591836735
t = -6.281107509 + 4.591836735

Combine like terms: -6.281107509 + 4.591836735 = -1.689270774
t = -1.689270774

Simplifying
t = -1.689270774


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {10.872944244, -1.689270774}

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