15+13t(84-t)=84

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Solution for 15+13t(84-t)=84 equation:


Simplifying
15 + 13t(84 + -1t) = 84
15 + (84 * 13t + -1t * 13t) = 84
15 + (1092t + -13t2) = 84

Solving
15 + 1092t + -13t2 = 84

Solving for variable 't'.

Reorder the terms:
15 + -84 + 1092t + -13t2 = 84 + -84

Combine like terms: 15 + -84 = -69
-69 + 1092t + -13t2 = 84 + -84

Combine like terms: 84 + -84 = 0
-69 + 1092t + -13t2 = 0

Begin completing the square.  Divide all terms by
-13 the coefficient of the squared term: 

Divide each side by '-13'.
5.307692308 + -84t + t2 = 0

Move the constant term to the right:

Add '-5.307692308' to each side of the equation.
5.307692308 + -84t + -5.307692308 + t2 = 0 + -5.307692308

Reorder the terms:
5.307692308 + -5.307692308 + -84t + t2 = 0 + -5.307692308

Combine like terms: 5.307692308 + -5.307692308 = 0.000000000
0.000000000 + -84t + t2 = 0 + -5.307692308
-84t + t2 = 0 + -5.307692308

Combine like terms: 0 + -5.307692308 = -5.307692308
-84t + t2 = -5.307692308

The t term is -84t.  Take half its coefficient (-42).
Square it (1764) and add it to both sides.

Add '1764' to each side of the equation.
-84t + 1764 + t2 = -5.307692308 + 1764

Reorder the terms:
1764 + -84t + t2 = -5.307692308 + 1764

Combine like terms: -5.307692308 + 1764 = 1758.692307692
1764 + -84t + t2 = 1758.692307692

Factor a perfect square on the left side:
(t + -42)(t + -42) = 1758.692307692

Calculate the square root of the right side: 41.936765585

Break this problem into two subproblems by setting 
(t + -42) equal to 41.936765585 and -41.936765585.

Subproblem 1

t + -42 = 41.936765585 Simplifying t + -42 = 41.936765585 Reorder the terms: -42 + t = 41.936765585 Solving -42 + t = 41.936765585 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '42' to each side of the equation. -42 + 42 + t = 41.936765585 + 42 Combine like terms: -42 + 42 = 0 0 + t = 41.936765585 + 42 t = 41.936765585 + 42 Combine like terms: 41.936765585 + 42 = 83.936765585 t = 83.936765585 Simplifying t = 83.936765585

Subproblem 2

t + -42 = -41.936765585 Simplifying t + -42 = -41.936765585 Reorder the terms: -42 + t = -41.936765585 Solving -42 + t = -41.936765585 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '42' to each side of the equation. -42 + 42 + t = -41.936765585 + 42 Combine like terms: -42 + 42 = 0 0 + t = -41.936765585 + 42 t = -41.936765585 + 42 Combine like terms: -41.936765585 + 42 = 0.063234415 t = 0.063234415 Simplifying t = 0.063234415

Solution

The solution to the problem is based on the solutions from the subproblems. t = {83.936765585, 0.063234415}

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