t^2+27t+75=0

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Solution for t^2+27t+75=0 equation:


Simplifying
t2 + 27t + 75 = 0

Reorder the terms:
75 + 27t + t2 = 0

Solving
75 + 27t + t2 = 0

Solving for variable 't'.

Begin completing the square.

Move the constant term to the right:

Add '-75' to each side of the equation.
75 + 27t + -75 + t2 = 0 + -75

Reorder the terms:
75 + -75 + 27t + t2 = 0 + -75

Combine like terms: 75 + -75 = 0
0 + 27t + t2 = 0 + -75
27t + t2 = 0 + -75

Combine like terms: 0 + -75 = -75
27t + t2 = -75

The t term is 27t.  Take half its coefficient (13.5).
Square it (182.25) and add it to both sides.

Add '182.25' to each side of the equation.
27t + 182.25 + t2 = -75 + 182.25

Reorder the terms:
182.25 + 27t + t2 = -75 + 182.25

Combine like terms: -75 + 182.25 = 107.25
182.25 + 27t + t2 = 107.25

Factor a perfect square on the left side:
(t + 13.5)(t + 13.5) = 107.25

Calculate the square root of the right side: 10.356157589

Break this problem into two subproblems by setting 
(t + 13.5) equal to 10.356157589 and -10.356157589.

Subproblem 1

t + 13.5 = 10.356157589 Simplifying t + 13.5 = 10.356157589 Reorder the terms: 13.5 + t = 10.356157589 Solving 13.5 + t = 10.356157589 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-13.5' to each side of the equation. 13.5 + -13.5 + t = 10.356157589 + -13.5 Combine like terms: 13.5 + -13.5 = 0.0 0.0 + t = 10.356157589 + -13.5 t = 10.356157589 + -13.5 Combine like terms: 10.356157589 + -13.5 = -3.143842411 t = -3.143842411 Simplifying t = -3.143842411

Subproblem 2

t + 13.5 = -10.356157589 Simplifying t + 13.5 = -10.356157589 Reorder the terms: 13.5 + t = -10.356157589 Solving 13.5 + t = -10.356157589 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-13.5' to each side of the equation. 13.5 + -13.5 + t = -10.356157589 + -13.5 Combine like terms: 13.5 + -13.5 = 0.0 0.0 + t = -10.356157589 + -13.5 t = -10.356157589 + -13.5 Combine like terms: -10.356157589 + -13.5 = -23.856157589 t = -23.856157589 Simplifying t = -23.856157589

Solution

The solution to the problem is based on the solutions from the subproblems. t = {-3.143842411, -23.856157589}

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