53.1=3t+4.9t^2

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Solution for 53.1=3t+4.9t^2 equation:


Simplifying
53.1 = 3t + 4.9t2

Solving
53.1 = 3t + 4.9t2

Solving for variable 't'.

Reorder the terms:
53.1 + -3t + -4.9t2 = 3t + -3t + 4.9t2 + -4.9t2

Combine like terms: 3t + -3t = 0
53.1 + -3t + -4.9t2 = 0 + 4.9t2 + -4.9t2
53.1 + -3t + -4.9t2 = 4.9t2 + -4.9t2

Combine like terms: 4.9t2 + -4.9t2 = 0.0
53.1 + -3t + -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'.
-10.83673469 + 0.612244898t + t2 = 0

Move the constant term to the right:

Add '10.83673469' to each side of the equation.
-10.83673469 + 0.612244898t + 10.83673469 + t2 = 0 + 10.83673469

Reorder the terms:
-10.83673469 + 10.83673469 + 0.612244898t + t2 = 0 + 10.83673469

Combine like terms: -10.83673469 + 10.83673469 = 0.00000000
0.00000000 + 0.612244898t + t2 = 0 + 10.83673469
0.612244898t + t2 = 0 + 10.83673469

Combine like terms: 0 + 10.83673469 = 10.83673469
0.612244898t + t2 = 10.83673469

The t term is 0.612244898t.  Take half its coefficient (0.306122449).
Square it (0.09371095378) and add it to both sides.

Add '0.09371095378' to each side of the equation.
0.612244898t + 0.09371095378 + t2 = 10.83673469 + 0.09371095378

Reorder the terms:
0.09371095378 + 0.612244898t + t2 = 10.83673469 + 0.09371095378

Combine like terms: 10.83673469 + 0.09371095378 = 10.93044564378
0.09371095378 + 0.612244898t + t2 = 10.93044564378

Factor a perfect square on the left side:
(t + 0.306122449)(t + 0.306122449) = 10.93044564378

Calculate the square root of the right side: 3.306122448

Break this problem into two subproblems by setting 
(t + 0.306122449) equal to 3.306122448 and -3.306122448.

Subproblem 1

t + 0.306122449 = 3.306122448 Simplifying t + 0.306122449 = 3.306122448 Reorder the terms: 0.306122449 + t = 3.306122448 Solving 0.306122449 + t = 3.306122448 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.306122449' to each side of the equation. 0.306122449 + -0.306122449 + t = 3.306122448 + -0.306122449 Combine like terms: 0.306122449 + -0.306122449 = 0.000000000 0.000000000 + t = 3.306122448 + -0.306122449 t = 3.306122448 + -0.306122449 Combine like terms: 3.306122448 + -0.306122449 = 2.999999999 t = 2.999999999 Simplifying t = 2.999999999

Subproblem 2

t + 0.306122449 = -3.306122448 Simplifying t + 0.306122449 = -3.306122448 Reorder the terms: 0.306122449 + t = -3.306122448 Solving 0.306122449 + t = -3.306122448 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.306122449' to each side of the equation. 0.306122449 + -0.306122449 + t = -3.306122448 + -0.306122449 Combine like terms: 0.306122449 + -0.306122449 = 0.000000000 0.000000000 + t = -3.306122448 + -0.306122449 t = -3.306122448 + -0.306122449 Combine like terms: -3.306122448 + -0.306122449 = -3.612244897 t = -3.612244897 Simplifying t = -3.612244897

Solution

The solution to the problem is based on the solutions from the subproblems. t = {2.999999999, -3.612244897}

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