13(t-5)5t=6(3t+5)-10

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Solution for 13(t-5)5t=6(3t+5)-10 equation:


Simplifying
13(t + -5) * 5t = 6(3t + 5) + -10

Reorder the terms:
13(-5 + t) * 5t = 6(3t + 5) + -10

Reorder the terms for easier multiplication:
13 * 5t(-5 + t) = 6(3t + 5) + -10

Multiply 13 * 5
65t(-5 + t) = 6(3t + 5) + -10
(-5 * 65t + t * 65t) = 6(3t + 5) + -10
(-325t + 65t2) = 6(3t + 5) + -10

Reorder the terms:
-325t + 65t2 = 6(5 + 3t) + -10
-325t + 65t2 = (5 * 6 + 3t * 6) + -10
-325t + 65t2 = (30 + 18t) + -10

Reorder the terms:
-325t + 65t2 = 30 + -10 + 18t

Combine like terms: 30 + -10 = 20
-325t + 65t2 = 20 + 18t

Solving
-325t + 65t2 = 20 + 18t

Solving for variable 't'.

Reorder the terms:
-20 + -325t + -18t + 65t2 = 20 + 18t + -20 + -18t

Combine like terms: -325t + -18t = -343t
-20 + -343t + 65t2 = 20 + 18t + -20 + -18t

Reorder the terms:
-20 + -343t + 65t2 = 20 + -20 + 18t + -18t

Combine like terms: 20 + -20 = 0
-20 + -343t + 65t2 = 0 + 18t + -18t
-20 + -343t + 65t2 = 18t + -18t

Combine like terms: 18t + -18t = 0
-20 + -343t + 65t2 = 0

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

Divide each side by '65'.
-0.3076923077 + -5.276923077t + t2 = 0

Move the constant term to the right:

Add '0.3076923077' to each side of the equation.
-0.3076923077 + -5.276923077t + 0.3076923077 + t2 = 0 + 0.3076923077

Reorder the terms:
-0.3076923077 + 0.3076923077 + -5.276923077t + t2 = 0 + 0.3076923077

Combine like terms: -0.3076923077 + 0.3076923077 = 0.0000000000
0.0000000000 + -5.276923077t + t2 = 0 + 0.3076923077
-5.276923077t + t2 = 0 + 0.3076923077

Combine like terms: 0 + 0.3076923077 = 0.3076923077
-5.276923077t + t2 = 0.3076923077

The t term is -5.276923077t.  Take half its coefficient (-2.638461539).
Square it (6.961479293) and add it to both sides.

Add '6.961479293' to each side of the equation.
-5.276923077t + 6.961479293 + t2 = 0.3076923077 + 6.961479293

Reorder the terms:
6.961479293 + -5.276923077t + t2 = 0.3076923077 + 6.961479293

Combine like terms: 0.3076923077 + 6.961479293 = 7.2691716007
6.961479293 + -5.276923077t + t2 = 7.2691716007

Factor a perfect square on the left side:
(t + -2.638461539)(t + -2.638461539) = 7.2691716007

Calculate the square root of the right side: 2.69614013

Break this problem into two subproblems by setting 
(t + -2.638461539) equal to 2.69614013 and -2.69614013.

Subproblem 1

t + -2.638461539 = 2.69614013 Simplifying t + -2.638461539 = 2.69614013 Reorder the terms: -2.638461539 + t = 2.69614013 Solving -2.638461539 + t = 2.69614013 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.638461539' to each side of the equation. -2.638461539 + 2.638461539 + t = 2.69614013 + 2.638461539 Combine like terms: -2.638461539 + 2.638461539 = 0.000000000 0.000000000 + t = 2.69614013 + 2.638461539 t = 2.69614013 + 2.638461539 Combine like terms: 2.69614013 + 2.638461539 = 5.334601669 t = 5.334601669 Simplifying t = 5.334601669

Subproblem 2

t + -2.638461539 = -2.69614013 Simplifying t + -2.638461539 = -2.69614013 Reorder the terms: -2.638461539 + t = -2.69614013 Solving -2.638461539 + t = -2.69614013 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.638461539' to each side of the equation. -2.638461539 + 2.638461539 + t = -2.69614013 + 2.638461539 Combine like terms: -2.638461539 + 2.638461539 = 0.000000000 0.000000000 + t = -2.69614013 + 2.638461539 t = -2.69614013 + 2.638461539 Combine like terms: -2.69614013 + 2.638461539 = -0.057678591 t = -0.057678591 Simplifying t = -0.057678591

Solution

The solution to the problem is based on the solutions from the subproblems. t = {5.334601669, -0.057678591}

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