y^2-13y=40

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Solution for y^2-13y=40 equation:


Simplifying
y2 + -13y = 40

Reorder the terms:
-13y + y2 = 40

Solving
-13y + y2 = 40

Solving for variable 'y'.

Reorder the terms:
-40 + -13y + y2 = 40 + -40

Combine like terms: 40 + -40 = 0
-40 + -13y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + -13y + 40 + y2 = 0 + 40

Reorder the terms:
-40 + 40 + -13y + y2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + -13y + y2 = 0 + 40
-13y + y2 = 0 + 40

Combine like terms: 0 + 40 = 40
-13y + y2 = 40

The y term is -13y.  Take half its coefficient (-6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
-13y + 42.25 + y2 = 40 + 42.25

Reorder the terms:
42.25 + -13y + y2 = 40 + 42.25

Combine like terms: 40 + 42.25 = 82.25
42.25 + -13y + y2 = 82.25

Factor a perfect square on the left side:
(y + -6.5)(y + -6.5) = 82.25

Calculate the square root of the right side: 9.069178574

Break this problem into two subproblems by setting 
(y + -6.5) equal to 9.069178574 and -9.069178574.

Subproblem 1

y + -6.5 = 9.069178574 Simplifying y + -6.5 = 9.069178574 Reorder the terms: -6.5 + y = 9.069178574 Solving -6.5 + y = 9.069178574 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + y = 9.069178574 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = 9.069178574 + 6.5 y = 9.069178574 + 6.5 Combine like terms: 9.069178574 + 6.5 = 15.569178574 y = 15.569178574 Simplifying y = 15.569178574

Subproblem 2

y + -6.5 = -9.069178574 Simplifying y + -6.5 = -9.069178574 Reorder the terms: -6.5 + y = -9.069178574 Solving -6.5 + y = -9.069178574 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + y = -9.069178574 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + y = -9.069178574 + 6.5 y = -9.069178574 + 6.5 Combine like terms: -9.069178574 + 6.5 = -2.569178574 y = -2.569178574 Simplifying y = -2.569178574

Solution

The solution to the problem is based on the solutions from the subproblems. y = {15.569178574, -2.569178574}

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