16t^2+54t+7=0

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


Simplifying
16t2 + 54t + 7 = 0

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

Solving
7 + 54t + 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'.
0.4375 + 3.375t + t2 = 0

Move the constant term to the right:

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

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

Combine like terms: 0.4375 + -0.4375 = 0.0000
0.0000 + 3.375t + t2 = 0 + -0.4375
3.375t + t2 = 0 + -0.4375

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

The t term is 3.375t.  Take half its coefficient (1.6875).
Square it (2.84765625) and add it to both sides.

Add '2.84765625' to each side of the equation.
3.375t + 2.84765625 + t2 = -0.4375 + 2.84765625

Reorder the terms:
2.84765625 + 3.375t + t2 = -0.4375 + 2.84765625

Combine like terms: -0.4375 + 2.84765625 = 2.41015625
2.84765625 + 3.375t + t2 = 2.41015625

Factor a perfect square on the left side:
(t + 1.6875)(t + 1.6875) = 2.41015625

Calculate the square root of the right side: 1.552467794

Break this problem into two subproblems by setting 
(t + 1.6875) equal to 1.552467794 and -1.552467794.

Subproblem 1

t + 1.6875 = 1.552467794 Simplifying t + 1.6875 = 1.552467794 Reorder the terms: 1.6875 + t = 1.552467794 Solving 1.6875 + t = 1.552467794 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.6875' to each side of the equation. 1.6875 + -1.6875 + t = 1.552467794 + -1.6875 Combine like terms: 1.6875 + -1.6875 = 0.0000 0.0000 + t = 1.552467794 + -1.6875 t = 1.552467794 + -1.6875 Combine like terms: 1.552467794 + -1.6875 = -0.135032206 t = -0.135032206 Simplifying t = -0.135032206

Subproblem 2

t + 1.6875 = -1.552467794 Simplifying t + 1.6875 = -1.552467794 Reorder the terms: 1.6875 + t = -1.552467794 Solving 1.6875 + t = -1.552467794 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.6875' to each side of the equation. 1.6875 + -1.6875 + t = -1.552467794 + -1.6875 Combine like terms: 1.6875 + -1.6875 = 0.0000 0.0000 + t = -1.552467794 + -1.6875 t = -1.552467794 + -1.6875 Combine like terms: -1.552467794 + -1.6875 = -3.239967794 t = -3.239967794 Simplifying t = -3.239967794

Solution

The solution to the problem is based on the solutions from the subproblems. t = {-0.135032206, -3.239967794}

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