10t(t-4)+4t=7(2t+2)-13

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Solution for 10t(t-4)+4t=7(2t+2)-13 equation:


Simplifying
10t(t + -4) + 4t = 7(2t + 2) + -13

Reorder the terms:
10t(-4 + t) + 4t = 7(2t + 2) + -13
(-4 * 10t + t * 10t) + 4t = 7(2t + 2) + -13
(-40t + 10t2) + 4t = 7(2t + 2) + -13

Reorder the terms:
-40t + 4t + 10t2 = 7(2t + 2) + -13

Combine like terms: -40t + 4t = -36t
-36t + 10t2 = 7(2t + 2) + -13

Reorder the terms:
-36t + 10t2 = 7(2 + 2t) + -13
-36t + 10t2 = (2 * 7 + 2t * 7) + -13
-36t + 10t2 = (14 + 14t) + -13

Reorder the terms:
-36t + 10t2 = 14 + -13 + 14t

Combine like terms: 14 + -13 = 1
-36t + 10t2 = 1 + 14t

Solving
-36t + 10t2 = 1 + 14t

Solving for variable 't'.

Reorder the terms:
-1 + -36t + -14t + 10t2 = 1 + 14t + -1 + -14t

Combine like terms: -36t + -14t = -50t
-1 + -50t + 10t2 = 1 + 14t + -1 + -14t

Reorder the terms:
-1 + -50t + 10t2 = 1 + -1 + 14t + -14t

Combine like terms: 1 + -1 = 0
-1 + -50t + 10t2 = 0 + 14t + -14t
-1 + -50t + 10t2 = 14t + -14t

Combine like terms: 14t + -14t = 0
-1 + -50t + 10t2 = 0

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

Divide each side by '10'.
-0.1 + -5t + t2 = 0

Move the constant term to the right:

Add '0.1' to each side of the equation.
-0.1 + -5t + 0.1 + t2 = 0 + 0.1

Reorder the terms:
-0.1 + 0.1 + -5t + t2 = 0 + 0.1

Combine like terms: -0.1 + 0.1 = 0.0
0.0 + -5t + t2 = 0 + 0.1
-5t + t2 = 0 + 0.1

Combine like terms: 0 + 0.1 = 0.1
-5t + t2 = 0.1

The t term is -5t.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5t + 6.25 + t2 = 0.1 + 6.25

Reorder the terms:
6.25 + -5t + t2 = 0.1 + 6.25

Combine like terms: 0.1 + 6.25 = 6.35
6.25 + -5t + t2 = 6.35

Factor a perfect square on the left side:
(t + -2.5)(t + -2.5) = 6.35

Calculate the square root of the right side: 2.519920634

Break this problem into two subproblems by setting 
(t + -2.5) equal to 2.519920634 and -2.519920634.

Subproblem 1

t + -2.5 = 2.519920634 Simplifying t + -2.5 = 2.519920634 Reorder the terms: -2.5 + t = 2.519920634 Solving -2.5 + t = 2.519920634 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = 2.519920634 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 2.519920634 + 2.5 t = 2.519920634 + 2.5 Combine like terms: 2.519920634 + 2.5 = 5.019920634 t = 5.019920634 Simplifying t = 5.019920634

Subproblem 2

t + -2.5 = -2.519920634 Simplifying t + -2.5 = -2.519920634 Reorder the terms: -2.5 + t = -2.519920634 Solving -2.5 + t = -2.519920634 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = -2.519920634 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -2.519920634 + 2.5 t = -2.519920634 + 2.5 Combine like terms: -2.519920634 + 2.5 = -0.019920634 t = -0.019920634 Simplifying t = -0.019920634

Solution

The solution to the problem is based on the solutions from the subproblems. t = {5.019920634, -0.019920634}

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