312=128t-t^2

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Solution for 312=128t-t^2 equation:


Simplifying
312 = 128t + -1t2

Solving
312 = 128t + -1t2

Solving for variable 't'.

Reorder the terms:
312 + -128t + t2 = 128t + -128t + -1t2 + t2

Combine like terms: 128t + -128t = 0
312 + -128t + t2 = 0 + -1t2 + t2
312 + -128t + t2 = -1t2 + t2

Combine like terms: -1t2 + t2 = 0
312 + -128t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-312' to each side of the equation.
312 + -128t + -312 + t2 = 0 + -312

Reorder the terms:
312 + -312 + -128t + t2 = 0 + -312

Combine like terms: 312 + -312 = 0
0 + -128t + t2 = 0 + -312
-128t + t2 = 0 + -312

Combine like terms: 0 + -312 = -312
-128t + t2 = -312

The t term is -128t.  Take half its coefficient (-64).
Square it (4096) and add it to both sides.

Add '4096' to each side of the equation.
-128t + 4096 + t2 = -312 + 4096

Reorder the terms:
4096 + -128t + t2 = -312 + 4096

Combine like terms: -312 + 4096 = 3784
4096 + -128t + t2 = 3784

Factor a perfect square on the left side:
(t + -64)(t + -64) = 3784

Calculate the square root of the right side: 61.514225997

Break this problem into two subproblems by setting 
(t + -64) equal to 61.514225997 and -61.514225997.

Subproblem 1

t + -64 = 61.514225997 Simplifying t + -64 = 61.514225997 Reorder the terms: -64 + t = 61.514225997 Solving -64 + t = 61.514225997 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '64' to each side of the equation. -64 + 64 + t = 61.514225997 + 64 Combine like terms: -64 + 64 = 0 0 + t = 61.514225997 + 64 t = 61.514225997 + 64 Combine like terms: 61.514225997 + 64 = 125.514225997 t = 125.514225997 Simplifying t = 125.514225997

Subproblem 2

t + -64 = -61.514225997 Simplifying t + -64 = -61.514225997 Reorder the terms: -64 + t = -61.514225997 Solving -64 + t = -61.514225997 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '64' to each side of the equation. -64 + 64 + t = -61.514225997 + 64 Combine like terms: -64 + 64 = 0 0 + t = -61.514225997 + 64 t = -61.514225997 + 64 Combine like terms: -61.514225997 + 64 = 2.485774003 t = 2.485774003 Simplifying t = 2.485774003

Solution

The solution to the problem is based on the solutions from the subproblems. t = {125.514225997, 2.485774003}

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