-45=10t-4.9t^2

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Solution for -45=10t-4.9t^2 equation:


Simplifying
-45 = 10t + -4.9t2

Solving
-45 = 10t + -4.9t2

Solving for variable 't'.

Reorder the terms:
-45 + -10t + 4.9t2 = 10t + -10t + -4.9t2 + 4.9t2

Combine like terms: 10t + -10t = 0
-45 + -10t + 4.9t2 = 0 + -4.9t2 + 4.9t2
-45 + -10t + 4.9t2 = -4.9t2 + 4.9t2

Combine like terms: -4.9t2 + 4.9t2 = 0.0
-45 + -10t + 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'.
-9.183673469 + -2.040816327t + t2 = 0

Move the constant term to the right:

Add '9.183673469' to each side of the equation.
-9.183673469 + -2.040816327t + 9.183673469 + t2 = 0 + 9.183673469

Reorder the terms:
-9.183673469 + 9.183673469 + -2.040816327t + t2 = 0 + 9.183673469

Combine like terms: -9.183673469 + 9.183673469 = 0.000000000
0.000000000 + -2.040816327t + t2 = 0 + 9.183673469
-2.040816327t + t2 = 0 + 9.183673469

Combine like terms: 0 + 9.183673469 = 9.183673469
-2.040816327t + t2 = 9.183673469

The t term is -2.040816327t.  Take half its coefficient (-1.020408164).
Square it (1.041232821) and add it to both sides.

Add '1.041232821' to each side of the equation.
-2.040816327t + 1.041232821 + t2 = 9.183673469 + 1.041232821

Reorder the terms:
1.041232821 + -2.040816327t + t2 = 9.183673469 + 1.041232821

Combine like terms: 9.183673469 + 1.041232821 = 10.22490629
1.041232821 + -2.040816327t + t2 = 10.22490629

Factor a perfect square on the left side:
(t + -1.020408164)(t + -1.020408164) = 10.22490629

Calculate the square root of the right side: 3.197640738

Break this problem into two subproblems by setting 
(t + -1.020408164) equal to 3.197640738 and -3.197640738.

Subproblem 1

t + -1.020408164 = 3.197640738 Simplifying t + -1.020408164 = 3.197640738 Reorder the terms: -1.020408164 + t = 3.197640738 Solving -1.020408164 + t = 3.197640738 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.020408164' to each side of the equation. -1.020408164 + 1.020408164 + t = 3.197640738 + 1.020408164 Combine like terms: -1.020408164 + 1.020408164 = 0.000000000 0.000000000 + t = 3.197640738 + 1.020408164 t = 3.197640738 + 1.020408164 Combine like terms: 3.197640738 + 1.020408164 = 4.218048902 t = 4.218048902 Simplifying t = 4.218048902

Subproblem 2

t + -1.020408164 = -3.197640738 Simplifying t + -1.020408164 = -3.197640738 Reorder the terms: -1.020408164 + t = -3.197640738 Solving -1.020408164 + t = -3.197640738 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.020408164' to each side of the equation. -1.020408164 + 1.020408164 + t = -3.197640738 + 1.020408164 Combine like terms: -1.020408164 + 1.020408164 = 0.000000000 0.000000000 + t = -3.197640738 + 1.020408164 t = -3.197640738 + 1.020408164 Combine like terms: -3.197640738 + 1.020408164 = -2.177232574 t = -2.177232574 Simplifying t = -2.177232574

Solution

The solution to the problem is based on the solutions from the subproblems. t = {4.218048902, -2.177232574}

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