0=-4.9t^2+23.8t+3.5

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Solution for 0=-4.9t^2+23.8t+3.5 equation:


Simplifying
0 = -4.9t2 + 23.8t + 3.5

Reorder the terms:
0 = 3.5 + 23.8t + -4.9t2

Solving
0 = 3.5 + 23.8t + -4.9t2

Solving for variable 't'.

Combine like terms: 0 + -3.5 = -3.5
-3.5 + -23.8t + 4.9t2 = 3.5 + 23.8t + -4.9t2 + -3.5 + -23.8t + 4.9t2

Reorder the terms:
-3.5 + -23.8t + 4.9t2 = 3.5 + -3.5 + 23.8t + -23.8t + -4.9t2 + 4.9t2

Combine like terms: 3.5 + -3.5 = 0.0
-3.5 + -23.8t + 4.9t2 = 0.0 + 23.8t + -23.8t + -4.9t2 + 4.9t2
-3.5 + -23.8t + 4.9t2 = 23.8t + -23.8t + -4.9t2 + 4.9t2

Combine like terms: 23.8t + -23.8t = 0.0
-3.5 + -23.8t + 4.9t2 = 0.0 + -4.9t2 + 4.9t2
-3.5 + -23.8t + 4.9t2 = -4.9t2 + 4.9t2

Combine like terms: -4.9t2 + 4.9t2 = 0.0
-3.5 + -23.8t + 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'.
-0.7142857143 + -4.857142857t + t2 = 0

Move the constant term to the right:

Add '0.7142857143' to each side of the equation.
-0.7142857143 + -4.857142857t + 0.7142857143 + t2 = 0 + 0.7142857143

Reorder the terms:
-0.7142857143 + 0.7142857143 + -4.857142857t + t2 = 0 + 0.7142857143

Combine like terms: -0.7142857143 + 0.7142857143 = 0.0000000000
0.0000000000 + -4.857142857t + t2 = 0 + 0.7142857143
-4.857142857t + t2 = 0 + 0.7142857143

Combine like terms: 0 + 0.7142857143 = 0.7142857143
-4.857142857t + t2 = 0.7142857143

The t term is -4.857142857t.  Take half its coefficient (-2.428571429).
Square it (5.897959186) and add it to both sides.

Add '5.897959186' to each side of the equation.
-4.857142857t + 5.897959186 + t2 = 0.7142857143 + 5.897959186

Reorder the terms:
5.897959186 + -4.857142857t + t2 = 0.7142857143 + 5.897959186

Combine like terms: 0.7142857143 + 5.897959186 = 6.6122449003
5.897959186 + -4.857142857t + t2 = 6.6122449003

Factor a perfect square on the left side:
(t + -2.428571429)(t + -2.428571429) = 6.6122449003

Calculate the square root of the right side: 2.571428572

Break this problem into two subproblems by setting 
(t + -2.428571429) equal to 2.571428572 and -2.571428572.

Subproblem 1

t + -2.428571429 = 2.571428572 Simplifying t + -2.428571429 = 2.571428572 Reorder the terms: -2.428571429 + t = 2.571428572 Solving -2.428571429 + t = 2.571428572 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.428571429' to each side of the equation. -2.428571429 + 2.428571429 + t = 2.571428572 + 2.428571429 Combine like terms: -2.428571429 + 2.428571429 = 0.000000000 0.000000000 + t = 2.571428572 + 2.428571429 t = 2.571428572 + 2.428571429 Combine like terms: 2.571428572 + 2.428571429 = 5.000000001 t = 5.000000001 Simplifying t = 5.000000001

Subproblem 2

t + -2.428571429 = -2.571428572 Simplifying t + -2.428571429 = -2.571428572 Reorder the terms: -2.428571429 + t = -2.571428572 Solving -2.428571429 + t = -2.571428572 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.428571429' to each side of the equation. -2.428571429 + 2.428571429 + t = -2.571428572 + 2.428571429 Combine like terms: -2.428571429 + 2.428571429 = 0.000000000 0.000000000 + t = -2.571428572 + 2.428571429 t = -2.571428572 + 2.428571429 Combine like terms: -2.571428572 + 2.428571429 = -0.142857143 t = -0.142857143 Simplifying t = -0.142857143

Solution

The solution to the problem is based on the solutions from the subproblems. t = {5.000000001, -0.142857143}

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