16t^2-72t-3=84

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Solution for 16t^2-72t-3=84 equation:


Simplifying
16t2 + -72t + -3 = 84

Reorder the terms:
-3 + -72t + 16t2 = 84

Solving
-3 + -72t + 16t2 = 84

Solving for variable 't'.

Reorder the terms:
-3 + -84 + -72t + 16t2 = 84 + -84

Combine like terms: -3 + -84 = -87
-87 + -72t + 16t2 = 84 + -84

Combine like terms: 84 + -84 = 0
-87 + -72t + 16t2 = 0

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

Divide each side by '16'.
-5.4375 + -4.5t + t2 = 0

Move the constant term to the right:

Add '5.4375' to each side of the equation.
-5.4375 + -4.5t + 5.4375 + t2 = 0 + 5.4375

Reorder the terms:
-5.4375 + 5.4375 + -4.5t + t2 = 0 + 5.4375

Combine like terms: -5.4375 + 5.4375 = 0.0000
0.0000 + -4.5t + t2 = 0 + 5.4375
-4.5t + t2 = 0 + 5.4375

Combine like terms: 0 + 5.4375 = 5.4375
-4.5t + t2 = 5.4375

The t term is -4.5t.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5t + 5.0625 + t2 = 5.4375 + 5.0625

Reorder the terms:
5.0625 + -4.5t + t2 = 5.4375 + 5.0625

Combine like terms: 5.4375 + 5.0625 = 10.5
5.0625 + -4.5t + t2 = 10.5

Factor a perfect square on the left side:
(t + -2.25)(t + -2.25) = 10.5

Calculate the square root of the right side: 3.240370349

Break this problem into two subproblems by setting 
(t + -2.25) equal to 3.240370349 and -3.240370349.

Subproblem 1

t + -2.25 = 3.240370349 Simplifying t + -2.25 = 3.240370349 Reorder the terms: -2.25 + t = 3.240370349 Solving -2.25 + t = 3.240370349 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + t = 3.240370349 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + t = 3.240370349 + 2.25 t = 3.240370349 + 2.25 Combine like terms: 3.240370349 + 2.25 = 5.490370349 t = 5.490370349 Simplifying t = 5.490370349

Subproblem 2

t + -2.25 = -3.240370349 Simplifying t + -2.25 = -3.240370349 Reorder the terms: -2.25 + t = -3.240370349 Solving -2.25 + t = -3.240370349 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + t = -3.240370349 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + t = -3.240370349 + 2.25 t = -3.240370349 + 2.25 Combine like terms: -3.240370349 + 2.25 = -0.990370349 t = -0.990370349 Simplifying t = -0.990370349

Solution

The solution to the problem is based on the solutions from the subproblems. t = {5.490370349, -0.990370349}

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