3(3t-4)-(t^2+5)-4t(t-3)=0

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Solution for 3(3t-4)-(t^2+5)-4t(t-3)=0 equation:


Simplifying
3(3t + -4) + -1(t2 + 5) + -4t(t + -3) = 0

Reorder the terms:
3(-4 + 3t) + -1(t2 + 5) + -4t(t + -3) = 0
(-4 * 3 + 3t * 3) + -1(t2 + 5) + -4t(t + -3) = 0
(-12 + 9t) + -1(t2 + 5) + -4t(t + -3) = 0

Reorder the terms:
-12 + 9t + -1(5 + t2) + -4t(t + -3) = 0
-12 + 9t + (5 * -1 + t2 * -1) + -4t(t + -3) = 0
-12 + 9t + (-5 + -1t2) + -4t(t + -3) = 0

Reorder the terms:
-12 + 9t + -5 + -1t2 + -4t(-3 + t) = 0
-12 + 9t + -5 + -1t2 + (-3 * -4t + t * -4t) = 0
-12 + 9t + -5 + -1t2 + (12t + -4t2) = 0

Reorder the terms:
-12 + -5 + 9t + 12t + -1t2 + -4t2 = 0

Combine like terms: -12 + -5 = -17
-17 + 9t + 12t + -1t2 + -4t2 = 0

Combine like terms: 9t + 12t = 21t
-17 + 21t + -1t2 + -4t2 = 0

Combine like terms: -1t2 + -4t2 = -5t2
-17 + 21t + -5t2 = 0

Solving
-17 + 21t + -5t2 = 0

Solving for variable 't'.

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
3.4 + -4.2t + t2 = 0

Move the constant term to the right:

Add '-3.4' to each side of the equation.
3.4 + -4.2t + -3.4 + t2 = 0 + -3.4

Reorder the terms:
3.4 + -3.4 + -4.2t + t2 = 0 + -3.4

Combine like terms: 3.4 + -3.4 = 0.0
0.0 + -4.2t + t2 = 0 + -3.4
-4.2t + t2 = 0 + -3.4

Combine like terms: 0 + -3.4 = -3.4
-4.2t + t2 = -3.4

The t term is -4.2t.  Take half its coefficient (-2.1).
Square it (4.41) and add it to both sides.

Add '4.41' to each side of the equation.
-4.2t + 4.41 + t2 = -3.4 + 4.41

Reorder the terms:
4.41 + -4.2t + t2 = -3.4 + 4.41

Combine like terms: -3.4 + 4.41 = 1.01
4.41 + -4.2t + t2 = 1.01

Factor a perfect square on the left side:
(t + -2.1)(t + -2.1) = 1.01

Calculate the square root of the right side: 1.004987562

Break this problem into two subproblems by setting 
(t + -2.1) equal to 1.004987562 and -1.004987562.

Subproblem 1

t + -2.1 = 1.004987562 Simplifying t + -2.1 = 1.004987562 Reorder the terms: -2.1 + t = 1.004987562 Solving -2.1 + t = 1.004987562 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.1' to each side of the equation. -2.1 + 2.1 + t = 1.004987562 + 2.1 Combine like terms: -2.1 + 2.1 = 0.0 0.0 + t = 1.004987562 + 2.1 t = 1.004987562 + 2.1 Combine like terms: 1.004987562 + 2.1 = 3.104987562 t = 3.104987562 Simplifying t = 3.104987562

Subproblem 2

t + -2.1 = -1.004987562 Simplifying t + -2.1 = -1.004987562 Reorder the terms: -2.1 + t = -1.004987562 Solving -2.1 + t = -1.004987562 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.1' to each side of the equation. -2.1 + 2.1 + t = -1.004987562 + 2.1 Combine like terms: -2.1 + 2.1 = 0.0 0.0 + t = -1.004987562 + 2.1 t = -1.004987562 + 2.1 Combine like terms: -1.004987562 + 2.1 = 1.095012438 t = 1.095012438 Simplifying t = 1.095012438

Solution

The solution to the problem is based on the solutions from the subproblems. t = {3.104987562, 1.095012438}

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