2t(t+3)+3=0

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Solution for 2t(t+3)+3=0 equation:


Simplifying
2t(t + 3) + 3 = 0

Reorder the terms:
2t(3 + t) + 3 = 0
(3 * 2t + t * 2t) + 3 = 0
(6t + 2t2) + 3 = 0

Reorder the terms:
3 + 6t + 2t2 = 0

Solving
3 + 6t + 2t2 = 0

Solving for variable 't'.

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

Divide each side by '2'.
1.5 + 3t + t2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 3t + -1.5 + t2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 3t + t2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 3t + t2 = 0 + -1.5
3t + t2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
3t + t2 = -1.5

The t term is 3t.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3t + 2.25 + t2 = -1.5 + 2.25

Reorder the terms:
2.25 + 3t + t2 = -1.5 + 2.25

Combine like terms: -1.5 + 2.25 = 0.75
2.25 + 3t + t2 = 0.75

Factor a perfect square on the left side:
(t + 1.5)(t + 1.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(t + 1.5) equal to 0.866025404 and -0.866025404.

Subproblem 1

t + 1.5 = 0.866025404 Simplifying t + 1.5 = 0.866025404 Reorder the terms: 1.5 + t = 0.866025404 Solving 1.5 + t = 0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = 0.866025404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = 0.866025404 + -1.5 t = 0.866025404 + -1.5 Combine like terms: 0.866025404 + -1.5 = -0.633974596 t = -0.633974596 Simplifying t = -0.633974596

Subproblem 2

t + 1.5 = -0.866025404 Simplifying t + 1.5 = -0.866025404 Reorder the terms: 1.5 + t = -0.866025404 Solving 1.5 + t = -0.866025404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = -0.866025404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = -0.866025404 + -1.5 t = -0.866025404 + -1.5 Combine like terms: -0.866025404 + -1.5 = -2.366025404 t = -2.366025404 Simplifying t = -2.366025404

Solution

The solution to the problem is based on the solutions from the subproblems. t = {-0.633974596, -2.366025404}

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