35=5t+3t^2

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Solution for 35=5t+3t^2 equation:


Simplifying
35 = 5t + 3t2

Solving
35 = 5t + 3t2

Solving for variable 't'.

Reorder the terms:
35 + -5t + -3t2 = 5t + -5t + 3t2 + -3t2

Combine like terms: 5t + -5t = 0
35 + -5t + -3t2 = 0 + 3t2 + -3t2
35 + -5t + -3t2 = 3t2 + -3t2

Combine like terms: 3t2 + -3t2 = 0
35 + -5t + -3t2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-11.66666667 + 1.666666667t + t2 = 0

Move the constant term to the right:

Add '11.66666667' to each side of the equation.
-11.66666667 + 1.666666667t + 11.66666667 + t2 = 0 + 11.66666667

Reorder the terms:
-11.66666667 + 11.66666667 + 1.666666667t + t2 = 0 + 11.66666667

Combine like terms: -11.66666667 + 11.66666667 = 0.00000000
0.00000000 + 1.666666667t + t2 = 0 + 11.66666667
1.666666667t + t2 = 0 + 11.66666667

Combine like terms: 0 + 11.66666667 = 11.66666667
1.666666667t + t2 = 11.66666667

The t term is 1.666666667t.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667t + 0.6944444447 + t2 = 11.66666667 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667t + t2 = 11.66666667 + 0.6944444447

Combine like terms: 11.66666667 + 0.6944444447 = 12.3611111147
0.6944444447 + 1.666666667t + t2 = 12.3611111147

Factor a perfect square on the left side:
(t + 0.8333333335)(t + 0.8333333335) = 12.3611111147

Calculate the square root of the right side: 3.515837185

Break this problem into two subproblems by setting 
(t + 0.8333333335) equal to 3.515837185 and -3.515837185.

Subproblem 1

t + 0.8333333335 = 3.515837185 Simplifying t + 0.8333333335 = 3.515837185 Reorder the terms: 0.8333333335 + t = 3.515837185 Solving 0.8333333335 + t = 3.515837185 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + t = 3.515837185 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + t = 3.515837185 + -0.8333333335 t = 3.515837185 + -0.8333333335 Combine like terms: 3.515837185 + -0.8333333335 = 2.6825038515 t = 2.6825038515 Simplifying t = 2.6825038515

Subproblem 2

t + 0.8333333335 = -3.515837185 Simplifying t + 0.8333333335 = -3.515837185 Reorder the terms: 0.8333333335 + t = -3.515837185 Solving 0.8333333335 + t = -3.515837185 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + t = -3.515837185 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + t = -3.515837185 + -0.8333333335 t = -3.515837185 + -0.8333333335 Combine like terms: -3.515837185 + -0.8333333335 = -4.3491705185 t = -4.3491705185 Simplifying t = -4.3491705185

Solution

The solution to the problem is based on the solutions from the subproblems. t = {2.6825038515, -4.3491705185}

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