t(2t-3)=40

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Solution for t(2t-3)=40 equation:


Simplifying
t(2t + -3) = 40

Reorder the terms:
t(-3 + 2t) = 40
(-3 * t + 2t * t) = 40
(-3t + 2t2) = 40

Solving
-3t + 2t2 = 40

Solving for variable 't'.

Reorder the terms:
-40 + -3t + 2t2 = 40 + -40

Combine like terms: 40 + -40 = 0
-40 + -3t + 2t2 = 0

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

Divide each side by '2'.
-20 + -1.5t + t2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + -1.5t + 20 + t2 = 0 + 20

Reorder the terms:
-20 + 20 + -1.5t + t2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + -1.5t + t2 = 0 + 20
-1.5t + t2 = 0 + 20

Combine like terms: 0 + 20 = 20
-1.5t + t2 = 20

The t term is -1.5t.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5t + 0.5625 + t2 = 20 + 0.5625

Reorder the terms:
0.5625 + -1.5t + t2 = 20 + 0.5625

Combine like terms: 20 + 0.5625 = 20.5625
0.5625 + -1.5t + t2 = 20.5625

Factor a perfect square on the left side:
(t + -0.75)(t + -0.75) = 20.5625

Calculate the square root of the right side: 4.534589287

Break this problem into two subproblems by setting 
(t + -0.75) equal to 4.534589287 and -4.534589287.

Subproblem 1

t + -0.75 = 4.534589287 Simplifying t + -0.75 = 4.534589287 Reorder the terms: -0.75 + t = 4.534589287 Solving -0.75 + t = 4.534589287 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = 4.534589287 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 4.534589287 + 0.75 t = 4.534589287 + 0.75 Combine like terms: 4.534589287 + 0.75 = 5.284589287 t = 5.284589287 Simplifying t = 5.284589287

Subproblem 2

t + -0.75 = -4.534589287 Simplifying t + -0.75 = -4.534589287 Reorder the terms: -0.75 + t = -4.534589287 Solving -0.75 + t = -4.534589287 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = -4.534589287 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -4.534589287 + 0.75 t = -4.534589287 + 0.75 Combine like terms: -4.534589287 + 0.75 = -3.784589287 t = -3.784589287 Simplifying t = -3.784589287

Solution

The solution to the problem is based on the solutions from the subproblems. t = {5.284589287, -3.784589287}

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