4t^2+(14t(t+0.5))*0.5=24

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Solution for 4t^2+(14t(t+0.5))*0.5=24 equation:


Simplifying
4t2 + (14t(t + 0.5)) * 0.5 = 24

Reorder the terms:
4t2 + (14t(0.5 + t)) * 0.5 = 24
4t2 + ((0.5 * 14t + t * 14t)) * 0.5 = 24
4t2 + ((7t + 14t2)) * 0.5 = 24

Reorder the terms for easier multiplication:
4t2 + 0.5(7t + 14t2) = 24
4t2 + (7t * 0.5 + 14t2 * 0.5) = 24
4t2 + (3.5t + 7t2) = 24

Reorder the terms:
3.5t + 4t2 + 7t2 = 24

Combine like terms: 4t2 + 7t2 = 11t2
3.5t + 11t2 = 24

Solving
3.5t + 11t2 = 24

Solving for variable 't'.

Reorder the terms:
-24 + 3.5t + 11t2 = 24 + -24

Combine like terms: 24 + -24 = 0
-24 + 3.5t + 11t2 = 0

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

Divide each side by '11'.
-2.181818182 + 0.3181818182t + t2 = 0

Move the constant term to the right:

Add '2.181818182' to each side of the equation.
-2.181818182 + 0.3181818182t + 2.181818182 + t2 = 0 + 2.181818182

Reorder the terms:
-2.181818182 + 2.181818182 + 0.3181818182t + t2 = 0 + 2.181818182

Combine like terms: -2.181818182 + 2.181818182 = 0.000000000
0.000000000 + 0.3181818182t + t2 = 0 + 2.181818182
0.3181818182t + t2 = 0 + 2.181818182

Combine like terms: 0 + 2.181818182 = 2.181818182
0.3181818182t + t2 = 2.181818182

The t term is 0.3181818182t.  Take half its coefficient (0.1590909091).
Square it (0.02530991736) and add it to both sides.

Add '0.02530991736' to each side of the equation.
0.3181818182t + 0.02530991736 + t2 = 2.181818182 + 0.02530991736

Reorder the terms:
0.02530991736 + 0.3181818182t + t2 = 2.181818182 + 0.02530991736

Combine like terms: 2.181818182 + 0.02530991736 = 2.20712809936
0.02530991736 + 0.3181818182t + t2 = 2.20712809936

Factor a perfect square on the left side:
(t + 0.1590909091)(t + 0.1590909091) = 2.20712809936

Calculate the square root of the right side: 1.485640636

Break this problem into two subproblems by setting 
(t + 0.1590909091) equal to 1.485640636 and -1.485640636.

Subproblem 1

t + 0.1590909091 = 1.485640636 Simplifying t + 0.1590909091 = 1.485640636 Reorder the terms: 0.1590909091 + t = 1.485640636 Solving 0.1590909091 + t = 1.485640636 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1590909091' to each side of the equation. 0.1590909091 + -0.1590909091 + t = 1.485640636 + -0.1590909091 Combine like terms: 0.1590909091 + -0.1590909091 = 0.0000000000 0.0000000000 + t = 1.485640636 + -0.1590909091 t = 1.485640636 + -0.1590909091 Combine like terms: 1.485640636 + -0.1590909091 = 1.3265497269 t = 1.3265497269 Simplifying t = 1.3265497269

Subproblem 2

t + 0.1590909091 = -1.485640636 Simplifying t + 0.1590909091 = -1.485640636 Reorder the terms: 0.1590909091 + t = -1.485640636 Solving 0.1590909091 + t = -1.485640636 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1590909091' to each side of the equation. 0.1590909091 + -0.1590909091 + t = -1.485640636 + -0.1590909091 Combine like terms: 0.1590909091 + -0.1590909091 = 0.0000000000 0.0000000000 + t = -1.485640636 + -0.1590909091 t = -1.485640636 + -0.1590909091 Combine like terms: -1.485640636 + -0.1590909091 = -1.6447315451 t = -1.6447315451 Simplifying t = -1.6447315451

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.3265497269, -1.6447315451}

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