217=16t^2+32t

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Solution for 217=16t^2+32t equation:


Simplifying
217 = 16t2 + 32t

Reorder the terms:
217 = 32t + 16t2

Solving
217 = 32t + 16t2

Solving for variable 't'.

Reorder the terms:
217 + -32t + -16t2 = 32t + -32t + 16t2 + -16t2

Combine like terms: 32t + -32t = 0
217 + -32t + -16t2 = 0 + 16t2 + -16t2
217 + -32t + -16t2 = 16t2 + -16t2

Combine like terms: 16t2 + -16t2 = 0
217 + -32t + -16t2 = 0

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

Divide each side by '-16'.
-13.5625 + 2t + t2 = 0

Move the constant term to the right:

Add '13.5625' to each side of the equation.
-13.5625 + 2t + 13.5625 + t2 = 0 + 13.5625

Reorder the terms:
-13.5625 + 13.5625 + 2t + t2 = 0 + 13.5625

Combine like terms: -13.5625 + 13.5625 = 0.0000
0.0000 + 2t + t2 = 0 + 13.5625
2t + t2 = 0 + 13.5625

Combine like terms: 0 + 13.5625 = 13.5625
2t + t2 = 13.5625

The t term is 2t.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2t + 1 + t2 = 13.5625 + 1

Reorder the terms:
1 + 2t + t2 = 13.5625 + 1

Combine like terms: 13.5625 + 1 = 14.5625
1 + 2t + t2 = 14.5625

Factor a perfect square on the left side:
(t + 1)(t + 1) = 14.5625

Calculate the square root of the right side: 3.816084381

Break this problem into two subproblems by setting 
(t + 1) equal to 3.816084381 and -3.816084381.

Subproblem 1

t + 1 = 3.816084381 Simplifying t + 1 = 3.816084381 Reorder the terms: 1 + t = 3.816084381 Solving 1 + t = 3.816084381 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = 3.816084381 + -1 Combine like terms: 1 + -1 = 0 0 + t = 3.816084381 + -1 t = 3.816084381 + -1 Combine like terms: 3.816084381 + -1 = 2.816084381 t = 2.816084381 Simplifying t = 2.816084381

Subproblem 2

t + 1 = -3.816084381 Simplifying t + 1 = -3.816084381 Reorder the terms: 1 + t = -3.816084381 Solving 1 + t = -3.816084381 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = -3.816084381 + -1 Combine like terms: 1 + -1 = 0 0 + t = -3.816084381 + -1 t = -3.816084381 + -1 Combine like terms: -3.816084381 + -1 = -4.816084381 t = -4.816084381 Simplifying t = -4.816084381

Solution

The solution to the problem is based on the solutions from the subproblems. t = {2.816084381, -4.816084381}

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