-2t^2+30t+6.5=100

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Solution for -2t^2+30t+6.5=100 equation:


Simplifying
-2t2 + 30t + 6.5 = 100

Reorder the terms:
6.5 + 30t + -2t2 = 100

Solving
6.5 + 30t + -2t2 = 100

Solving for variable 't'.

Reorder the terms:
6.5 + -100 + 30t + -2t2 = 100 + -100

Combine like terms: 6.5 + -100 = -93.5
-93.5 + 30t + -2t2 = 100 + -100

Combine like terms: 100 + -100 = 0
-93.5 + 30t + -2t2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
46.75 + -15t + t2 = 0

Move the constant term to the right:

Add '-46.75' to each side of the equation.
46.75 + -15t + -46.75 + t2 = 0 + -46.75

Reorder the terms:
46.75 + -46.75 + -15t + t2 = 0 + -46.75

Combine like terms: 46.75 + -46.75 = 0.00
0.00 + -15t + t2 = 0 + -46.75
-15t + t2 = 0 + -46.75

Combine like terms: 0 + -46.75 = -46.75
-15t + t2 = -46.75

The t term is -15t.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15t + 56.25 + t2 = -46.75 + 56.25

Reorder the terms:
56.25 + -15t + t2 = -46.75 + 56.25

Combine like terms: -46.75 + 56.25 = 9.5
56.25 + -15t + t2 = 9.5

Factor a perfect square on the left side:
(t + -7.5)(t + -7.5) = 9.5

Calculate the square root of the right side: 3.082207001

Break this problem into two subproblems by setting 
(t + -7.5) equal to 3.082207001 and -3.082207001.

Subproblem 1

t + -7.5 = 3.082207001 Simplifying t + -7.5 = 3.082207001 Reorder the terms: -7.5 + t = 3.082207001 Solving -7.5 + t = 3.082207001 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + t = 3.082207001 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = 3.082207001 + 7.5 t = 3.082207001 + 7.5 Combine like terms: 3.082207001 + 7.5 = 10.582207001 t = 10.582207001 Simplifying t = 10.582207001

Subproblem 2

t + -7.5 = -3.082207001 Simplifying t + -7.5 = -3.082207001 Reorder the terms: -7.5 + t = -3.082207001 Solving -7.5 + t = -3.082207001 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + t = -3.082207001 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = -3.082207001 + 7.5 t = -3.082207001 + 7.5 Combine like terms: -3.082207001 + 7.5 = 4.417792999 t = 4.417792999 Simplifying t = 4.417792999

Solution

The solution to the problem is based on the solutions from the subproblems. t = {10.582207001, 4.417792999}

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