.25t^2-4t+135=1000

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Solution for .25t^2-4t+135=1000 equation:


Simplifying
0.25t2 + -4t + 135 = 1000

Reorder the terms:
135 + -4t + 0.25t2 = 1000

Solving
135 + -4t + 0.25t2 = 1000

Solving for variable 't'.

Reorder the terms:
135 + -1000 + -4t + 0.25t2 = 1000 + -1000

Combine like terms: 135 + -1000 = -865
-865 + -4t + 0.25t2 = 1000 + -1000

Combine like terms: 1000 + -1000 = 0
-865 + -4t + 0.25t2 = 0

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

Divide each side by '0.25'.
-3460 + -16t + t2 = 0

Move the constant term to the right:

Add '3460' to each side of the equation.
-3460 + -16t + 3460 + t2 = 0 + 3460

Reorder the terms:
-3460 + 3460 + -16t + t2 = 0 + 3460

Combine like terms: -3460 + 3460 = 0
0 + -16t + t2 = 0 + 3460
-16t + t2 = 0 + 3460

Combine like terms: 0 + 3460 = 3460
-16t + t2 = 3460

The t term is -16t.  Take half its coefficient (-8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
-16t + 64 + t2 = 3460 + 64

Reorder the terms:
64 + -16t + t2 = 3460 + 64

Combine like terms: 3460 + 64 = 3524
64 + -16t + t2 = 3524

Factor a perfect square on the left side:
(t + -8)(t + -8) = 3524

Calculate the square root of the right side: 59.363288319

Break this problem into two subproblems by setting 
(t + -8) equal to 59.363288319 and -59.363288319.

Subproblem 1

t + -8 = 59.363288319 Simplifying t + -8 = 59.363288319 Reorder the terms: -8 + t = 59.363288319 Solving -8 + t = 59.363288319 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + t = 59.363288319 + 8 Combine like terms: -8 + 8 = 0 0 + t = 59.363288319 + 8 t = 59.363288319 + 8 Combine like terms: 59.363288319 + 8 = 67.363288319 t = 67.363288319 Simplifying t = 67.363288319

Subproblem 2

t + -8 = -59.363288319 Simplifying t + -8 = -59.363288319 Reorder the terms: -8 + t = -59.363288319 Solving -8 + t = -59.363288319 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + t = -59.363288319 + 8 Combine like terms: -8 + 8 = 0 0 + t = -59.363288319 + 8 t = -59.363288319 + 8 Combine like terms: -59.363288319 + 8 = -51.363288319 t = -51.363288319 Simplifying t = -51.363288319

Solution

The solution to the problem is based on the solutions from the subproblems. t = {67.363288319, -51.363288319}

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