.5t^2+12t-1000=0

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Solution for .5t^2+12t-1000=0 equation:


Simplifying
0.5t2 + 12t + -1000 = 0

Reorder the terms:
-1000 + 12t + 0.5t2 = 0

Solving
-1000 + 12t + 0.5t2 = 0

Solving for variable 't'.

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

Divide each side by '0.5'.
-2000 + 24t + t2 = 0

Move the constant term to the right:

Add '2000' to each side of the equation.
-2000 + 24t + 2000 + t2 = 0 + 2000

Reorder the terms:
-2000 + 2000 + 24t + t2 = 0 + 2000

Combine like terms: -2000 + 2000 = 0
0 + 24t + t2 = 0 + 2000
24t + t2 = 0 + 2000

Combine like terms: 0 + 2000 = 2000
24t + t2 = 2000

The t term is 24t.  Take half its coefficient (12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
24t + 144 + t2 = 2000 + 144

Reorder the terms:
144 + 24t + t2 = 2000 + 144

Combine like terms: 2000 + 144 = 2144
144 + 24t + t2 = 2144

Factor a perfect square on the left side:
(t + 12)(t + 12) = 2144

Calculate the square root of the right side: 46.303347611

Break this problem into two subproblems by setting 
(t + 12) equal to 46.303347611 and -46.303347611.

Subproblem 1

t + 12 = 46.303347611 Simplifying t + 12 = 46.303347611 Reorder the terms: 12 + t = 46.303347611 Solving 12 + t = 46.303347611 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + t = 46.303347611 + -12 Combine like terms: 12 + -12 = 0 0 + t = 46.303347611 + -12 t = 46.303347611 + -12 Combine like terms: 46.303347611 + -12 = 34.303347611 t = 34.303347611 Simplifying t = 34.303347611

Subproblem 2

t + 12 = -46.303347611 Simplifying t + 12 = -46.303347611 Reorder the terms: 12 + t = -46.303347611 Solving 12 + t = -46.303347611 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + t = -46.303347611 + -12 Combine like terms: 12 + -12 = 0 0 + t = -46.303347611 + -12 t = -46.303347611 + -12 Combine like terms: -46.303347611 + -12 = -58.303347611 t = -58.303347611 Simplifying t = -58.303347611

Solution

The solution to the problem is based on the solutions from the subproblems. t = {34.303347611, -58.303347611}

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