(2t-3)(2t-3)-2=24

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Solution for (2t-3)(2t-3)-2=24 equation:


Simplifying
(2t + -3)(2t + -3) + -2 = 24

Reorder the terms:
(-3 + 2t)(2t + -3) + -2 = 24

Reorder the terms:
(-3 + 2t)(-3 + 2t) + -2 = 24

Multiply (-3 + 2t) * (-3 + 2t)
(-3(-3 + 2t) + 2t * (-3 + 2t)) + -2 = 24
((-3 * -3 + 2t * -3) + 2t * (-3 + 2t)) + -2 = 24
((9 + -6t) + 2t * (-3 + 2t)) + -2 = 24
(9 + -6t + (-3 * 2t + 2t * 2t)) + -2 = 24
(9 + -6t + (-6t + 4t2)) + -2 = 24

Combine like terms: -6t + -6t = -12t
(9 + -12t + 4t2) + -2 = 24

Reorder the terms:
9 + -2 + -12t + 4t2 = 24

Combine like terms: 9 + -2 = 7
7 + -12t + 4t2 = 24

Solving
7 + -12t + 4t2 = 24

Solving for variable 't'.

Reorder the terms:
7 + -24 + -12t + 4t2 = 24 + -24

Combine like terms: 7 + -24 = -17
-17 + -12t + 4t2 = 24 + -24

Combine like terms: 24 + -24 = 0
-17 + -12t + 4t2 = 0

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

Divide each side by '4'.
-4.25 + -3t + t2 = 0

Move the constant term to the right:

Add '4.25' to each side of the equation.
-4.25 + -3t + 4.25 + t2 = 0 + 4.25

Reorder the terms:
-4.25 + 4.25 + -3t + t2 = 0 + 4.25

Combine like terms: -4.25 + 4.25 = 0.00
0.00 + -3t + t2 = 0 + 4.25
-3t + t2 = 0 + 4.25

Combine like terms: 0 + 4.25 = 4.25
-3t + t2 = 4.25

The t term is -3t.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3t + 2.25 + t2 = 4.25 + 2.25

Reorder the terms:
2.25 + -3t + t2 = 4.25 + 2.25

Combine like terms: 4.25 + 2.25 = 6.5
2.25 + -3t + t2 = 6.5

Factor a perfect square on the left side:
(t + -1.5)(t + -1.5) = 6.5

Calculate the square root of the right side: 2.549509757

Break this problem into two subproblems by setting 
(t + -1.5) equal to 2.549509757 and -2.549509757.

Subproblem 1

t + -1.5 = 2.549509757 Simplifying t + -1.5 = 2.549509757 Reorder the terms: -1.5 + t = 2.549509757 Solving -1.5 + t = 2.549509757 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = 2.549509757 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = 2.549509757 + 1.5 t = 2.549509757 + 1.5 Combine like terms: 2.549509757 + 1.5 = 4.049509757 t = 4.049509757 Simplifying t = 4.049509757

Subproblem 2

t + -1.5 = -2.549509757 Simplifying t + -1.5 = -2.549509757 Reorder the terms: -1.5 + t = -2.549509757 Solving -1.5 + t = -2.549509757 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + t = -2.549509757 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + t = -2.549509757 + 1.5 t = -2.549509757 + 1.5 Combine like terms: -2.549509757 + 1.5 = -1.049509757 t = -1.049509757 Simplifying t = -1.049509757

Solution

The solution to the problem is based on the solutions from the subproblems. t = {4.049509757, -1.049509757}

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