3t^2-8t-37=0

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Solution for 3t^2-8t-37=0 equation:


Simplifying
3t2 + -8t + -37 = 0

Reorder the terms:
-37 + -8t + 3t2 = 0

Solving
-37 + -8t + 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'.
-12.33333333 + -2.666666667t + t2 = 0

Move the constant term to the right:

Add '12.33333333' to each side of the equation.
-12.33333333 + -2.666666667t + 12.33333333 + t2 = 0 + 12.33333333

Reorder the terms:
-12.33333333 + 12.33333333 + -2.666666667t + t2 = 0 + 12.33333333

Combine like terms: -12.33333333 + 12.33333333 = 0.00000000
0.00000000 + -2.666666667t + t2 = 0 + 12.33333333
-2.666666667t + t2 = 0 + 12.33333333

Combine like terms: 0 + 12.33333333 = 12.33333333
-2.666666667t + t2 = 12.33333333

The t term is -2.666666667t.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667t + 1.777777780 + t2 = 12.33333333 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667t + t2 = 12.33333333 + 1.777777780

Combine like terms: 12.33333333 + 1.777777780 = 14.11111111
1.777777780 + -2.666666667t + t2 = 14.11111111

Factor a perfect square on the left side:
(t + -1.333333334)(t + -1.333333334) = 14.11111111

Calculate the square root of the right side: 3.75647589

Break this problem into two subproblems by setting 
(t + -1.333333334) equal to 3.75647589 and -3.75647589.

Subproblem 1

t + -1.333333334 = 3.75647589 Simplifying t + -1.333333334 = 3.75647589 Reorder the terms: -1.333333334 + t = 3.75647589 Solving -1.333333334 + t = 3.75647589 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + t = 3.75647589 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = 3.75647589 + 1.333333334 t = 3.75647589 + 1.333333334 Combine like terms: 3.75647589 + 1.333333334 = 5.089809224 t = 5.089809224 Simplifying t = 5.089809224

Subproblem 2

t + -1.333333334 = -3.75647589 Simplifying t + -1.333333334 = -3.75647589 Reorder the terms: -1.333333334 + t = -3.75647589 Solving -1.333333334 + t = -3.75647589 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + t = -3.75647589 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = -3.75647589 + 1.333333334 t = -3.75647589 + 1.333333334 Combine like terms: -3.75647589 + 1.333333334 = -2.423142556 t = -2.423142556 Simplifying t = -2.423142556

Solution

The solution to the problem is based on the solutions from the subproblems. t = {5.089809224, -2.423142556}

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