3t^2-4t-24=0

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Solution for 3t^2-4t-24=0 equation:


Simplifying
3t2 + -4t + -24 = 0

Reorder the terms:
-24 + -4t + 3t2 = 0

Solving
-24 + -4t + 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'.
-8 + -1.333333333t + t2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -1.333333333t + 8 + t2 = 0 + 8

Reorder the terms:
-8 + 8 + -1.333333333t + t2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -1.333333333t + t2 = 0 + 8
-1.333333333t + t2 = 0 + 8

Combine like terms: 0 + 8 = 8
-1.333333333t + t2 = 8

The t term is -1.333333333t.  Take half its coefficient (-0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
-1.333333333t + 0.4444444442 + t2 = 8 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333t + t2 = 8 + 0.4444444442

Combine like terms: 8 + 0.4444444442 = 8.4444444442
0.4444444442 + -1.333333333t + t2 = 8.4444444442

Factor a perfect square on the left side:
(t + -0.6666666665)(t + -0.6666666665) = 8.4444444442

Calculate the square root of the right side: 2.905932629

Break this problem into two subproblems by setting 
(t + -0.6666666665) equal to 2.905932629 and -2.905932629.

Subproblem 1

t + -0.6666666665 = 2.905932629 Simplifying t + -0.6666666665 = 2.905932629 Reorder the terms: -0.6666666665 + t = 2.905932629 Solving -0.6666666665 + t = 2.905932629 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + t = 2.905932629 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + t = 2.905932629 + 0.6666666665 t = 2.905932629 + 0.6666666665 Combine like terms: 2.905932629 + 0.6666666665 = 3.5725992955 t = 3.5725992955 Simplifying t = 3.5725992955

Subproblem 2

t + -0.6666666665 = -2.905932629 Simplifying t + -0.6666666665 = -2.905932629 Reorder the terms: -0.6666666665 + t = -2.905932629 Solving -0.6666666665 + t = -2.905932629 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + t = -2.905932629 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + t = -2.905932629 + 0.6666666665 t = -2.905932629 + 0.6666666665 Combine like terms: -2.905932629 + 0.6666666665 = -2.2392659625 t = -2.2392659625 Simplifying t = -2.2392659625

Solution

The solution to the problem is based on the solutions from the subproblems. t = {3.5725992955, -2.2392659625}

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