2t^2-3t-19=0

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Solution for 2t^2-3t-19=0 equation:


Simplifying
2t2 + -3t + -19 = 0

Reorder the terms:
-19 + -3t + 2t2 = 0

Solving
-19 + -3t + 2t2 = 0

Solving for variable 't'.

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

Divide each side by '2'.
-9.5 + -1.5t + t2 = 0

Move the constant term to the right:

Add '9.5' to each side of the equation.
-9.5 + -1.5t + 9.5 + t2 = 0 + 9.5

Reorder the terms:
-9.5 + 9.5 + -1.5t + t2 = 0 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + -1.5t + t2 = 0 + 9.5
-1.5t + t2 = 0 + 9.5

Combine like terms: 0 + 9.5 = 9.5
-1.5t + t2 = 9.5

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 = 9.5 + 0.5625

Reorder the terms:
0.5625 + -1.5t + t2 = 9.5 + 0.5625

Combine like terms: 9.5 + 0.5625 = 10.0625
0.5625 + -1.5t + t2 = 10.0625

Factor a perfect square on the left side:
(t + -0.75)(t + -0.75) = 10.0625

Calculate the square root of the right side: 3.172144385

Break this problem into two subproblems by setting 
(t + -0.75) equal to 3.172144385 and -3.172144385.

Subproblem 1

t + -0.75 = 3.172144385 Simplifying t + -0.75 = 3.172144385 Reorder the terms: -0.75 + t = 3.172144385 Solving -0.75 + t = 3.172144385 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 = 3.172144385 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 3.172144385 + 0.75 t = 3.172144385 + 0.75 Combine like terms: 3.172144385 + 0.75 = 3.922144385 t = 3.922144385 Simplifying t = 3.922144385

Subproblem 2

t + -0.75 = -3.172144385 Simplifying t + -0.75 = -3.172144385 Reorder the terms: -0.75 + t = -3.172144385 Solving -0.75 + t = -3.172144385 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 = -3.172144385 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -3.172144385 + 0.75 t = -3.172144385 + 0.75 Combine like terms: -3.172144385 + 0.75 = -2.422144385 t = -2.422144385 Simplifying t = -2.422144385

Solution

The solution to the problem is based on the solutions from the subproblems. t = {3.922144385, -2.422144385}

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