0=4.9t^2+345t-2070

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Solution for 0=4.9t^2+345t-2070 equation:


Simplifying
0 = 4.9t2 + 345t + -2070

Reorder the terms:
0 = -2070 + 345t + 4.9t2

Solving
0 = -2070 + 345t + 4.9t2

Solving for variable 't'.

Combine like terms: 0 + 2070 = 2070
2070 + -345t + -4.9t2 = -2070 + 345t + 4.9t2 + 2070 + -345t + -4.9t2

Reorder the terms:
2070 + -345t + -4.9t2 = -2070 + 2070 + 345t + -345t + 4.9t2 + -4.9t2

Combine like terms: -2070 + 2070 = 0
2070 + -345t + -4.9t2 = 0 + 345t + -345t + 4.9t2 + -4.9t2
2070 + -345t + -4.9t2 = 345t + -345t + 4.9t2 + -4.9t2

Combine like terms: 345t + -345t = 0
2070 + -345t + -4.9t2 = 0 + 4.9t2 + -4.9t2
2070 + -345t + -4.9t2 = 4.9t2 + -4.9t2

Combine like terms: 4.9t2 + -4.9t2 = 0.0
2070 + -345t + -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'.
-422.4489796 + 70.40816327t + t2 = 0

Move the constant term to the right:

Add '422.4489796' to each side of the equation.
-422.4489796 + 70.40816327t + 422.4489796 + t2 = 0 + 422.4489796

Reorder the terms:
-422.4489796 + 422.4489796 + 70.40816327t + t2 = 0 + 422.4489796

Combine like terms: -422.4489796 + 422.4489796 = 0.0000000
0.0000000 + 70.40816327t + t2 = 0 + 422.4489796
70.40816327t + t2 = 0 + 422.4489796

Combine like terms: 0 + 422.4489796 = 422.4489796
70.40816327t + t2 = 422.4489796

The t term is 70.40816327t.  Take half its coefficient (35.20408164).
Square it (1239.327364) and add it to both sides.

Add '1239.327364' to each side of the equation.
70.40816327t + 1239.327364 + t2 = 422.4489796 + 1239.327364

Reorder the terms:
1239.327364 + 70.40816327t + t2 = 422.4489796 + 1239.327364

Combine like terms: 422.4489796 + 1239.327364 = 1661.7763436
1239.327364 + 70.40816327t + t2 = 1661.7763436

Factor a perfect square on the left side:
(t + 35.20408164)(t + 35.20408164) = 1661.7763436

Calculate the square root of the right side: 40.764891066

Break this problem into two subproblems by setting 
(t + 35.20408164) equal to 40.764891066 and -40.764891066.

Subproblem 1

t + 35.20408164 = 40.764891066 Simplifying t + 35.20408164 = 40.764891066 Reorder the terms: 35.20408164 + t = 40.764891066 Solving 35.20408164 + t = 40.764891066 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-35.20408164' to each side of the equation. 35.20408164 + -35.20408164 + t = 40.764891066 + -35.20408164 Combine like terms: 35.20408164 + -35.20408164 = 0.00000000 0.00000000 + t = 40.764891066 + -35.20408164 t = 40.764891066 + -35.20408164 Combine like terms: 40.764891066 + -35.20408164 = 5.560809426 t = 5.560809426 Simplifying t = 5.560809426

Subproblem 2

t + 35.20408164 = -40.764891066 Simplifying t + 35.20408164 = -40.764891066 Reorder the terms: 35.20408164 + t = -40.764891066 Solving 35.20408164 + t = -40.764891066 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-35.20408164' to each side of the equation. 35.20408164 + -35.20408164 + t = -40.764891066 + -35.20408164 Combine like terms: 35.20408164 + -35.20408164 = 0.00000000 0.00000000 + t = -40.764891066 + -35.20408164 t = -40.764891066 + -35.20408164 Combine like terms: -40.764891066 + -35.20408164 = -75.968972706 t = -75.968972706 Simplifying t = -75.968972706

Solution

The solution to the problem is based on the solutions from the subproblems. t = {5.560809426, -75.968972706}

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