180=-4.9t^2+67t

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Solution for 180=-4.9t^2+67t equation:


Simplifying
180 = -4.9t2 + 67t

Reorder the terms:
180 = 67t + -4.9t2

Solving
180 = 67t + -4.9t2

Solving for variable 't'.

Reorder the terms:
180 + -67t + 4.9t2 = 67t + -67t + -4.9t2 + 4.9t2

Combine like terms: 67t + -67t = 0
180 + -67t + 4.9t2 = 0 + -4.9t2 + 4.9t2
180 + -67t + 4.9t2 = -4.9t2 + 4.9t2

Combine like terms: -4.9t2 + 4.9t2 = 0.0
180 + -67t + 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'.
36.73469388 + -13.67346939t + t2 = 0

Move the constant term to the right:

Add '-36.73469388' to each side of the equation.
36.73469388 + -13.67346939t + -36.73469388 + t2 = 0 + -36.73469388

Reorder the terms:
36.73469388 + -36.73469388 + -13.67346939t + t2 = 0 + -36.73469388

Combine like terms: 36.73469388 + -36.73469388 = 0.00000000
0.00000000 + -13.67346939t + t2 = 0 + -36.73469388
-13.67346939t + t2 = 0 + -36.73469388

Combine like terms: 0 + -36.73469388 = -36.73469388
-13.67346939t + t2 = -36.73469388

The t term is -13.67346939t.  Take half its coefficient (-6.836734695).
Square it (46.74094129) and add it to both sides.

Add '46.74094129' to each side of the equation.
-13.67346939t + 46.74094129 + t2 = -36.73469388 + 46.74094129

Reorder the terms:
46.74094129 + -13.67346939t + t2 = -36.73469388 + 46.74094129

Combine like terms: -36.73469388 + 46.74094129 = 10.00624741
46.74094129 + -13.67346939t + t2 = 10.00624741

Factor a perfect square on the left side:
(t + -6.836734695)(t + -6.836734695) = 10.00624741

Calculate the square root of the right side: 3.163265308

Break this problem into two subproblems by setting 
(t + -6.836734695) equal to 3.163265308 and -3.163265308.

Subproblem 1

t + -6.836734695 = 3.163265308 Simplifying t + -6.836734695 = 3.163265308 Reorder the terms: -6.836734695 + t = 3.163265308 Solving -6.836734695 + t = 3.163265308 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6.836734695' to each side of the equation. -6.836734695 + 6.836734695 + t = 3.163265308 + 6.836734695 Combine like terms: -6.836734695 + 6.836734695 = 0.000000000 0.000000000 + t = 3.163265308 + 6.836734695 t = 3.163265308 + 6.836734695 Combine like terms: 3.163265308 + 6.836734695 = 10.000000003 t = 10.000000003 Simplifying t = 10.000000003

Subproblem 2

t + -6.836734695 = -3.163265308 Simplifying t + -6.836734695 = -3.163265308 Reorder the terms: -6.836734695 + t = -3.163265308 Solving -6.836734695 + t = -3.163265308 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6.836734695' to each side of the equation. -6.836734695 + 6.836734695 + t = -3.163265308 + 6.836734695 Combine like terms: -6.836734695 + 6.836734695 = 0.000000000 0.000000000 + t = -3.163265308 + 6.836734695 t = -3.163265308 + 6.836734695 Combine like terms: -3.163265308 + 6.836734695 = 3.673469387 t = 3.673469387 Simplifying t = 3.673469387

Solution

The solution to the problem is based on the solutions from the subproblems. t = {10.000000003, 3.673469387}

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