16t^2+7t-54=0

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Solution for 16t^2+7t-54=0 equation:


Simplifying
16t2 + 7t + -54 = 0

Reorder the terms:
-54 + 7t + 16t2 = 0

Solving
-54 + 7t + 16t2 = 0

Solving for variable 't'.

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

Divide each side by '16'.
-3.375 + 0.4375t + t2 = 0

Move the constant term to the right:

Add '3.375' to each side of the equation.
-3.375 + 0.4375t + 3.375 + t2 = 0 + 3.375

Reorder the terms:
-3.375 + 3.375 + 0.4375t + t2 = 0 + 3.375

Combine like terms: -3.375 + 3.375 = 0.000
0.000 + 0.4375t + t2 = 0 + 3.375
0.4375t + t2 = 0 + 3.375

Combine like terms: 0 + 3.375 = 3.375
0.4375t + t2 = 3.375

The t term is 0.4375t.  Take half its coefficient (0.21875).
Square it (0.0478515625) and add it to both sides.

Add '0.0478515625' to each side of the equation.
0.4375t + 0.0478515625 + t2 = 3.375 + 0.0478515625

Reorder the terms:
0.0478515625 + 0.4375t + t2 = 3.375 + 0.0478515625

Combine like terms: 3.375 + 0.0478515625 = 3.4228515625
0.0478515625 + 0.4375t + t2 = 3.4228515625

Factor a perfect square on the left side:
(t + 0.21875)(t + 0.21875) = 3.4228515625

Calculate the square root of the right side: 1.850095014

Break this problem into two subproblems by setting 
(t + 0.21875) equal to 1.850095014 and -1.850095014.

Subproblem 1

t + 0.21875 = 1.850095014 Simplifying t + 0.21875 = 1.850095014 Reorder the terms: 0.21875 + t = 1.850095014 Solving 0.21875 + t = 1.850095014 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.21875' to each side of the equation. 0.21875 + -0.21875 + t = 1.850095014 + -0.21875 Combine like terms: 0.21875 + -0.21875 = 0.00000 0.00000 + t = 1.850095014 + -0.21875 t = 1.850095014 + -0.21875 Combine like terms: 1.850095014 + -0.21875 = 1.631345014 t = 1.631345014 Simplifying t = 1.631345014

Subproblem 2

t + 0.21875 = -1.850095014 Simplifying t + 0.21875 = -1.850095014 Reorder the terms: 0.21875 + t = -1.850095014 Solving 0.21875 + t = -1.850095014 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.21875' to each side of the equation. 0.21875 + -0.21875 + t = -1.850095014 + -0.21875 Combine like terms: 0.21875 + -0.21875 = 0.00000 0.00000 + t = -1.850095014 + -0.21875 t = -1.850095014 + -0.21875 Combine like terms: -1.850095014 + -0.21875 = -2.068845014 t = -2.068845014 Simplifying t = -2.068845014

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.631345014, -2.068845014}

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