Y^2-3Y=15

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Solution for Y^2-3Y=15 equation:


Simplifying
Y2 + -3Y = 15

Reorder the terms:
-3Y + Y2 = 15

Solving
-3Y + Y2 = 15

Solving for variable 'Y'.

Reorder the terms:
-15 + -3Y + Y2 = 15 + -15

Combine like terms: 15 + -15 = 0
-15 + -3Y + Y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + -3Y + 15 + Y2 = 0 + 15

Reorder the terms:
-15 + 15 + -3Y + Y2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + -3Y + Y2 = 0 + 15
-3Y + Y2 = 0 + 15

Combine like terms: 0 + 15 = 15
-3Y + Y2 = 15

The Y term is -3Y.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3Y + 2.25 + Y2 = 15 + 2.25

Reorder the terms:
2.25 + -3Y + Y2 = 15 + 2.25

Combine like terms: 15 + 2.25 = 17.25
2.25 + -3Y + Y2 = 17.25

Factor a perfect square on the left side:
(Y + -1.5)(Y + -1.5) = 17.25

Calculate the square root of the right side: 4.153311931

Break this problem into two subproblems by setting 
(Y + -1.5) equal to 4.153311931 and -4.153311931.

Subproblem 1

Y + -1.5 = 4.153311931 Simplifying Y + -1.5 = 4.153311931 Reorder the terms: -1.5 + Y = 4.153311931 Solving -1.5 + Y = 4.153311931 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + Y = 4.153311931 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + Y = 4.153311931 + 1.5 Y = 4.153311931 + 1.5 Combine like terms: 4.153311931 + 1.5 = 5.653311931 Y = 5.653311931 Simplifying Y = 5.653311931

Subproblem 2

Y + -1.5 = -4.153311931 Simplifying Y + -1.5 = -4.153311931 Reorder the terms: -1.5 + Y = -4.153311931 Solving -1.5 + Y = -4.153311931 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + Y = -4.153311931 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + Y = -4.153311931 + 1.5 Y = -4.153311931 + 1.5 Combine like terms: -4.153311931 + 1.5 = -2.653311931 Y = -2.653311931 Simplifying Y = -2.653311931

Solution

The solution to the problem is based on the solutions from the subproblems. Y = {5.653311931, -2.653311931}

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