3t^2-32t-96=0

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Solution for 3t^2-32t-96=0 equation:


Simplifying
3t2 + -32t + -96 = 0

Reorder the terms:
-96 + -32t + 3t2 = 0

Solving
-96 + -32t + 3t2 = 0

Solving for variable 't'.

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

Divide each side by '3'.
-32 + -10.66666667t + t2 = 0

Move the constant term to the right:

Add '32' to each side of the equation.
-32 + -10.66666667t + 32 + t2 = 0 + 32

Reorder the terms:
-32 + 32 + -10.66666667t + t2 = 0 + 32

Combine like terms: -32 + 32 = 0
0 + -10.66666667t + t2 = 0 + 32
-10.66666667t + t2 = 0 + 32

Combine like terms: 0 + 32 = 32
-10.66666667t + t2 = 32

The t term is -10.66666667t.  Take half its coefficient (-5.333333335).
Square it (28.44444446) and add it to both sides.

Add '28.44444446' to each side of the equation.
-10.66666667t + 28.44444446 + t2 = 32 + 28.44444446

Reorder the terms:
28.44444446 + -10.66666667t + t2 = 32 + 28.44444446

Combine like terms: 32 + 28.44444446 = 60.44444446
28.44444446 + -10.66666667t + t2 = 60.44444446

Factor a perfect square on the left side:
(t + -5.333333335)(t + -5.333333335) = 60.44444446

Calculate the square root of the right side: 7.774602527

Break this problem into two subproblems by setting 
(t + -5.333333335) equal to 7.774602527 and -7.774602527.

Subproblem 1

t + -5.333333335 = 7.774602527 Simplifying t + -5.333333335 = 7.774602527 Reorder the terms: -5.333333335 + t = 7.774602527 Solving -5.333333335 + t = 7.774602527 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.333333335' to each side of the equation. -5.333333335 + 5.333333335 + t = 7.774602527 + 5.333333335 Combine like terms: -5.333333335 + 5.333333335 = 0.000000000 0.000000000 + t = 7.774602527 + 5.333333335 t = 7.774602527 + 5.333333335 Combine like terms: 7.774602527 + 5.333333335 = 13.107935862 t = 13.107935862 Simplifying t = 13.107935862

Subproblem 2

t + -5.333333335 = -7.774602527 Simplifying t + -5.333333335 = -7.774602527 Reorder the terms: -5.333333335 + t = -7.774602527 Solving -5.333333335 + t = -7.774602527 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5.333333335' to each side of the equation. -5.333333335 + 5.333333335 + t = -7.774602527 + 5.333333335 Combine like terms: -5.333333335 + 5.333333335 = 0.000000000 0.000000000 + t = -7.774602527 + 5.333333335 t = -7.774602527 + 5.333333335 Combine like terms: -7.774602527 + 5.333333335 = -2.441269192 t = -2.441269192 Simplifying t = -2.441269192

Solution

The solution to the problem is based on the solutions from the subproblems. t = {13.107935862, -2.441269192}

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