3t(t-5)+10+2(t-2)=0

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Solution for 3t(t-5)+10+2(t-2)=0 equation:


Simplifying
3t(t + -5) + 10 + 2(t + -2) = 0

Reorder the terms:
3t(-5 + t) + 10 + 2(t + -2) = 0
(-5 * 3t + t * 3t) + 10 + 2(t + -2) = 0
(-15t + 3t2) + 10 + 2(t + -2) = 0

Reorder the terms:
-15t + 3t2 + 10 + 2(-2 + t) = 0
-15t + 3t2 + 10 + (-2 * 2 + t * 2) = 0
-15t + 3t2 + 10 + (-4 + 2t) = 0

Reorder the terms:
10 + -4 + -15t + 2t + 3t2 = 0

Combine like terms: 10 + -4 = 6
6 + -15t + 2t + 3t2 = 0

Combine like terms: -15t + 2t = -13t
6 + -13t + 3t2 = 0

Solving
6 + -13t + 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'.
2 + -4.333333333t + t2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -4.333333333t + -2 + t2 = 0 + -2

Reorder the terms:
2 + -2 + -4.333333333t + t2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -4.333333333t + t2 = 0 + -2
-4.333333333t + t2 = 0 + -2

Combine like terms: 0 + -2 = -2
-4.333333333t + t2 = -2

The t term is -4.333333333t.  Take half its coefficient (-2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
-4.333333333t + 4.694444446 + t2 = -2 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333t + t2 = -2 + 4.694444446

Combine like terms: -2 + 4.694444446 = 2.694444446
4.694444446 + -4.333333333t + t2 = 2.694444446

Factor a perfect square on the left side:
(t + -2.166666667)(t + -2.166666667) = 2.694444446

Calculate the square root of the right side: 1.641476301

Break this problem into two subproblems by setting 
(t + -2.166666667) equal to 1.641476301 and -1.641476301.

Subproblem 1

t + -2.166666667 = 1.641476301 Simplifying t + -2.166666667 = 1.641476301 Reorder the terms: -2.166666667 + t = 1.641476301 Solving -2.166666667 + t = 1.641476301 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.166666667' to each side of the equation. -2.166666667 + 2.166666667 + t = 1.641476301 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + t = 1.641476301 + 2.166666667 t = 1.641476301 + 2.166666667 Combine like terms: 1.641476301 + 2.166666667 = 3.808142968 t = 3.808142968 Simplifying t = 3.808142968

Subproblem 2

t + -2.166666667 = -1.641476301 Simplifying t + -2.166666667 = -1.641476301 Reorder the terms: -2.166666667 + t = -1.641476301 Solving -2.166666667 + t = -1.641476301 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.166666667' to each side of the equation. -2.166666667 + 2.166666667 + t = -1.641476301 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + t = -1.641476301 + 2.166666667 t = -1.641476301 + 2.166666667 Combine like terms: -1.641476301 + 2.166666667 = 0.525190366 t = 0.525190366 Simplifying t = 0.525190366

Solution

The solution to the problem is based on the solutions from the subproblems. t = {3.808142968, 0.525190366}

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