-2t^2+30t+6.5=12

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Solution for -2t^2+30t+6.5=12 equation:


Simplifying
-2t2 + 30t + 6.5 = 12

Reorder the terms:
6.5 + 30t + -2t2 = 12

Solving
6.5 + 30t + -2t2 = 12

Solving for variable 't'.

Reorder the terms:
6.5 + -12 + 30t + -2t2 = 12 + -12

Combine like terms: 6.5 + -12 = -5.5
-5.5 + 30t + -2t2 = 12 + -12

Combine like terms: 12 + -12 = 0
-5.5 + 30t + -2t2 = 0

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

Divide each side by '-2'.
2.75 + -15t + t2 = 0

Move the constant term to the right:

Add '-2.75' to each side of the equation.
2.75 + -15t + -2.75 + t2 = 0 + -2.75

Reorder the terms:
2.75 + -2.75 + -15t + t2 = 0 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + -15t + t2 = 0 + -2.75
-15t + t2 = 0 + -2.75

Combine like terms: 0 + -2.75 = -2.75
-15t + t2 = -2.75

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 = -2.75 + 56.25

Reorder the terms:
56.25 + -15t + t2 = -2.75 + 56.25

Combine like terms: -2.75 + 56.25 = 53.5
56.25 + -15t + t2 = 53.5

Factor a perfect square on the left side:
(t + -7.5)(t + -7.5) = 53.5

Calculate the square root of the right side: 7.314369419

Break this problem into two subproblems by setting 
(t + -7.5) equal to 7.314369419 and -7.314369419.

Subproblem 1

t + -7.5 = 7.314369419 Simplifying t + -7.5 = 7.314369419 Reorder the terms: -7.5 + t = 7.314369419 Solving -7.5 + t = 7.314369419 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 = 7.314369419 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = 7.314369419 + 7.5 t = 7.314369419 + 7.5 Combine like terms: 7.314369419 + 7.5 = 14.814369419 t = 14.814369419 Simplifying t = 14.814369419

Subproblem 2

t + -7.5 = -7.314369419 Simplifying t + -7.5 = -7.314369419 Reorder the terms: -7.5 + t = -7.314369419 Solving -7.5 + t = -7.314369419 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 = -7.314369419 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + t = -7.314369419 + 7.5 t = -7.314369419 + 7.5 Combine like terms: -7.314369419 + 7.5 = 0.185630581 t = 0.185630581 Simplifying t = 0.185630581

Solution

The solution to the problem is based on the solutions from the subproblems. t = {14.814369419, 0.185630581}

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