6t^2-5t-12=0

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Solution for 6t^2-5t-12=0 equation:


Simplifying
6t2 + -5t + -12 = 0

Reorder the terms:
-12 + -5t + 6t2 = 0

Solving
-12 + -5t + 6t2 = 0

Solving for variable 't'.

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

Divide each side by '6'.
-2 + -0.8333333333t + t2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -0.8333333333t + 2 + t2 = 0 + 2

Reorder the terms:
-2 + 2 + -0.8333333333t + t2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -0.8333333333t + t2 = 0 + 2
-0.8333333333t + t2 = 0 + 2

Combine like terms: 0 + 2 = 2
-0.8333333333t + t2 = 2

The t term is -0.8333333333t.  Take half its coefficient (-0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
-0.8333333333t + 0.1736111111 + t2 = 2 + 0.1736111111

Reorder the terms:
0.1736111111 + -0.8333333333t + t2 = 2 + 0.1736111111

Combine like terms: 2 + 0.1736111111 = 2.1736111111
0.1736111111 + -0.8333333333t + t2 = 2.1736111111

Factor a perfect square on the left side:
(t + -0.4166666667)(t + -0.4166666667) = 2.1736111111

Calculate the square root of the right side: 1.474317168

Break this problem into two subproblems by setting 
(t + -0.4166666667) equal to 1.474317168 and -1.474317168.

Subproblem 1

t + -0.4166666667 = 1.474317168 Simplifying t + -0.4166666667 = 1.474317168 Reorder the terms: -0.4166666667 + t = 1.474317168 Solving -0.4166666667 + t = 1.474317168 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + t = 1.474317168 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + t = 1.474317168 + 0.4166666667 t = 1.474317168 + 0.4166666667 Combine like terms: 1.474317168 + 0.4166666667 = 1.8909838347 t = 1.8909838347 Simplifying t = 1.8909838347

Subproblem 2

t + -0.4166666667 = -1.474317168 Simplifying t + -0.4166666667 = -1.474317168 Reorder the terms: -0.4166666667 + t = -1.474317168 Solving -0.4166666667 + t = -1.474317168 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + t = -1.474317168 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + t = -1.474317168 + 0.4166666667 t = -1.474317168 + 0.4166666667 Combine like terms: -1.474317168 + 0.4166666667 = -1.0576505013 t = -1.0576505013 Simplifying t = -1.0576505013

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.8909838347, -1.0576505013}

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