Y^2-Y-45=0

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Solution for Y^2-Y-45=0 equation:


Simplifying
Y2 + -1Y + -45 = 0

Reorder the terms:
-45 + -1Y + Y2 = 0

Solving
-45 + -1Y + Y2 = 0

Solving for variable 'Y'.

Begin completing the square.

Move the constant term to the right:

Add '45' to each side of the equation.
-45 + -1Y + 45 + Y2 = 0 + 45

Reorder the terms:
-45 + 45 + -1Y + Y2 = 0 + 45

Combine like terms: -45 + 45 = 0
0 + -1Y + Y2 = 0 + 45
-1Y + Y2 = 0 + 45

Combine like terms: 0 + 45 = 45
-1Y + Y2 = 45

The Y term is -1Y.  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.
-1Y + 0.25 + Y2 = 45 + 0.25

Reorder the terms:
0.25 + -1Y + Y2 = 45 + 0.25

Combine like terms: 45 + 0.25 = 45.25
0.25 + -1Y + Y2 = 45.25

Factor a perfect square on the left side:
(Y + -0.5)(Y + -0.5) = 45.25

Calculate the square root of the right side: 6.726812024

Break this problem into two subproblems by setting 
(Y + -0.5) equal to 6.726812024 and -6.726812024.

Subproblem 1

Y + -0.5 = 6.726812024 Simplifying Y + -0.5 = 6.726812024 Reorder the terms: -0.5 + Y = 6.726812024 Solving -0.5 + Y = 6.726812024 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + Y = 6.726812024 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + Y = 6.726812024 + 0.5 Y = 6.726812024 + 0.5 Combine like terms: 6.726812024 + 0.5 = 7.226812024 Y = 7.226812024 Simplifying Y = 7.226812024

Subproblem 2

Y + -0.5 = -6.726812024 Simplifying Y + -0.5 = -6.726812024 Reorder the terms: -0.5 + Y = -6.726812024 Solving -0.5 + Y = -6.726812024 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + Y = -6.726812024 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + Y = -6.726812024 + 0.5 Y = -6.726812024 + 0.5 Combine like terms: -6.726812024 + 0.5 = -6.226812024 Y = -6.226812024 Simplifying Y = -6.226812024

Solution

The solution to the problem is based on the solutions from the subproblems. Y = {7.226812024, -6.226812024}

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