2t(t+1)=7

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Solution for 2t(t+1)=7 equation:


Simplifying
2t(t + 1) = 7

Reorder the terms:
2t(1 + t) = 7
(1 * 2t + t * 2t) = 7
(2t + 2t2) = 7

Solving
2t + 2t2 = 7

Solving for variable 't'.

Reorder the terms:
-7 + 2t + 2t2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + 2t + 2t2 = 0

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

Divide each side by '2'.
-3.5 + t + t2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + t + 3.5 + t2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + t + t2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + t + t2 = 0 + 3.5
t + t2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
t + t2 = 3.5

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 = 3.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + t2 = 3.5 + 0.25

Combine like terms: 3.5 + 0.25 = 3.75
1.25 + t2 = 3.75

Factor a perfect square on the left side:
(t + 0.5)(t + 0.5) = 3.75

Calculate the square root of the right side: 1.936491673

Break this problem into two subproblems by setting 
(t + 0.5) equal to 1.936491673 and -1.936491673.

Subproblem 1

t + 0.5 = 1.936491673 Simplifying t + 0.5 = 1.936491673 Reorder the terms: 0.5 + t = 1.936491673 Solving 0.5 + t = 1.936491673 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 = 1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 1.936491673 + -0.5 t = 1.936491673 + -0.5 Combine like terms: 1.936491673 + -0.5 = 1.436491673 t = 1.436491673 Simplifying t = 1.436491673

Subproblem 2

t + 0.5 = -1.936491673 Simplifying t + 0.5 = -1.936491673 Reorder the terms: 0.5 + t = -1.936491673 Solving 0.5 + t = -1.936491673 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 = -1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -1.936491673 + -0.5 t = -1.936491673 + -0.5 Combine like terms: -1.936491673 + -0.5 = -2.436491673 t = -2.436491673 Simplifying t = -2.436491673

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.436491673, -2.436491673}

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