5(t^2-3)+2t=t(t-1)

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Solution for 5(t^2-3)+2t=t(t-1) equation:


Simplifying
5(t2 + -3) + 2t = t(t + -1)

Reorder the terms:
5(-3 + t2) + 2t = t(t + -1)
(-3 * 5 + t2 * 5) + 2t = t(t + -1)
(-15 + 5t2) + 2t = t(t + -1)

Reorder the terms:
-15 + 2t + 5t2 = t(t + -1)

Reorder the terms:
-15 + 2t + 5t2 = t(-1 + t)
-15 + 2t + 5t2 = (-1 * t + t * t)
-15 + 2t + 5t2 = (-1t + t2)

Solving
-15 + 2t + 5t2 = -1t + t2

Solving for variable 't'.

Reorder the terms:
-15 + 2t + t + 5t2 + -1t2 = -1t + t2 + t + -1t2

Combine like terms: 2t + t = 3t
-15 + 3t + 5t2 + -1t2 = -1t + t2 + t + -1t2

Combine like terms: 5t2 + -1t2 = 4t2
-15 + 3t + 4t2 = -1t + t2 + t + -1t2

Reorder the terms:
-15 + 3t + 4t2 = -1t + t + t2 + -1t2

Combine like terms: -1t + t = 0
-15 + 3t + 4t2 = 0 + t2 + -1t2
-15 + 3t + 4t2 = t2 + -1t2

Combine like terms: t2 + -1t2 = 0
-15 + 3t + 4t2 = 0

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

Divide each side by '4'.
-3.75 + 0.75t + t2 = 0

Move the constant term to the right:

Add '3.75' to each side of the equation.
-3.75 + 0.75t + 3.75 + t2 = 0 + 3.75

Reorder the terms:
-3.75 + 3.75 + 0.75t + t2 = 0 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + 0.75t + t2 = 0 + 3.75
0.75t + t2 = 0 + 3.75

Combine like terms: 0 + 3.75 = 3.75
0.75t + t2 = 3.75

The t term is 0.75t.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75t + 0.140625 + t2 = 3.75 + 0.140625

Reorder the terms:
0.140625 + 0.75t + t2 = 3.75 + 0.140625

Combine like terms: 3.75 + 0.140625 = 3.890625
0.140625 + 0.75t + t2 = 3.890625

Factor a perfect square on the left side:
(t + 0.375)(t + 0.375) = 3.890625

Calculate the square root of the right side: 1.97246673

Break this problem into two subproblems by setting 
(t + 0.375) equal to 1.97246673 and -1.97246673.

Subproblem 1

t + 0.375 = 1.97246673 Simplifying t + 0.375 = 1.97246673 Reorder the terms: 0.375 + t = 1.97246673 Solving 0.375 + t = 1.97246673 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = 1.97246673 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = 1.97246673 + -0.375 t = 1.97246673 + -0.375 Combine like terms: 1.97246673 + -0.375 = 1.59746673 t = 1.59746673 Simplifying t = 1.59746673

Subproblem 2

t + 0.375 = -1.97246673 Simplifying t + 0.375 = -1.97246673 Reorder the terms: 0.375 + t = -1.97246673 Solving 0.375 + t = -1.97246673 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = -1.97246673 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = -1.97246673 + -0.375 t = -1.97246673 + -0.375 Combine like terms: -1.97246673 + -0.375 = -2.34746673 t = -2.34746673 Simplifying t = -2.34746673

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.59746673, -2.34746673}

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