(2t+1)(2t+1)-0.25=0

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Solution for (2t+1)(2t+1)-0.25=0 equation:


Simplifying
(2t + 1)(2t + 1) + -0.25 = 0

Reorder the terms:
(1 + 2t)(2t + 1) + -0.25 = 0

Reorder the terms:
(1 + 2t)(1 + 2t) + -0.25 = 0

Multiply (1 + 2t) * (1 + 2t)
(1(1 + 2t) + 2t * (1 + 2t)) + -0.25 = 0
((1 * 1 + 2t * 1) + 2t * (1 + 2t)) + -0.25 = 0
((1 + 2t) + 2t * (1 + 2t)) + -0.25 = 0
(1 + 2t + (1 * 2t + 2t * 2t)) + -0.25 = 0
(1 + 2t + (2t + 4t2)) + -0.25 = 0

Combine like terms: 2t + 2t = 4t
(1 + 4t + 4t2) + -0.25 = 0

Reorder the terms:
1 + -0.25 + 4t + 4t2 = 0

Combine like terms: 1 + -0.25 = 0.75
0.75 + 4t + 4t2 = 0

Solving
0.75 + 4t + 4t2 = 0

Solving for variable 't'.

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

Divide each side by '4'.
0.1875 + t + t2 = 0

Move the constant term to the right:

Add '-0.1875' to each side of the equation.
0.1875 + t + -0.1875 + t2 = 0 + -0.1875

Reorder the terms:
0.1875 + -0.1875 + t + t2 = 0 + -0.1875

Combine like terms: 0.1875 + -0.1875 = 0.0000
0.0000 + t + t2 = 0 + -0.1875
t + t2 = 0 + -0.1875

Combine like terms: 0 + -0.1875 = -0.1875
t + t2 = -0.1875

The t term is t.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + t2 = -0.1875 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + t2 = -0.1875 + 0.25

Combine like terms: -0.1875 + 0.25 = 0.0625
1.25 + t2 = 0.0625

Factor a perfect square on the left side:
(t + 0.5)(t + 0.5) = 0.0625

Calculate the square root of the right side: 0.25

Break this problem into two subproblems by setting 
(t + 0.5) equal to 0.25 and -0.25.

Subproblem 1

t + 0.5 = 0.25 Simplifying t + 0.5 = 0.25 Reorder the terms: 0.5 + t = 0.25 Solving 0.5 + t = 0.25 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 0.25 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 0.25 + -0.5 t = 0.25 + -0.5 Combine like terms: 0.25 + -0.5 = -0.25 t = -0.25 Simplifying t = -0.25

Subproblem 2

t + 0.5 = -0.25 Simplifying t + 0.5 = -0.25 Reorder the terms: 0.5 + t = -0.25 Solving 0.5 + t = -0.25 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -0.25 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -0.25 + -0.5 t = -0.25 + -0.5 Combine like terms: -0.25 + -0.5 = -0.75 t = -0.75 Simplifying t = -0.75

Solution

The solution to the problem is based on the solutions from the subproblems. t = {-0.25, -0.75}

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