(y-3)(y-5)=18

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Solution for (y-3)(y-5)=18 equation:


Simplifying
(y + -3)(y + -5) = 18

Reorder the terms:
(-3 + y)(y + -5) = 18

Reorder the terms:
(-3 + y)(-5 + y) = 18

Multiply (-3 + y) * (-5 + y)
(-3(-5 + y) + y(-5 + y)) = 18
((-5 * -3 + y * -3) + y(-5 + y)) = 18
((15 + -3y) + y(-5 + y)) = 18
(15 + -3y + (-5 * y + y * y)) = 18
(15 + -3y + (-5y + y2)) = 18

Combine like terms: -3y + -5y = -8y
(15 + -8y + y2) = 18

Solving
15 + -8y + y2 = 18

Solving for variable 'y'.

Reorder the terms:
15 + -18 + -8y + y2 = 18 + -18

Combine like terms: 15 + -18 = -3
-3 + -8y + y2 = 18 + -18

Combine like terms: 18 + -18 = 0
-3 + -8y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -8y + 3 + y2 = 0 + 3

Reorder the terms:
-3 + 3 + -8y + y2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -8y + y2 = 0 + 3
-8y + y2 = 0 + 3

Combine like terms: 0 + 3 = 3
-8y + y2 = 3

The y term is -8y.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8y + 16 + y2 = 3 + 16

Reorder the terms:
16 + -8y + y2 = 3 + 16

Combine like terms: 3 + 16 = 19
16 + -8y + y2 = 19

Factor a perfect square on the left side:
(y + -4)(y + -4) = 19

Calculate the square root of the right side: 4.358898944

Break this problem into two subproblems by setting 
(y + -4) equal to 4.358898944 and -4.358898944.

Subproblem 1

y + -4 = 4.358898944 Simplifying y + -4 = 4.358898944 Reorder the terms: -4 + y = 4.358898944 Solving -4 + y = 4.358898944 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + y = 4.358898944 + 4 Combine like terms: -4 + 4 = 0 0 + y = 4.358898944 + 4 y = 4.358898944 + 4 Combine like terms: 4.358898944 + 4 = 8.358898944 y = 8.358898944 Simplifying y = 8.358898944

Subproblem 2

y + -4 = -4.358898944 Simplifying y + -4 = -4.358898944 Reorder the terms: -4 + y = -4.358898944 Solving -4 + y = -4.358898944 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + y = -4.358898944 + 4 Combine like terms: -4 + 4 = 0 0 + y = -4.358898944 + 4 y = -4.358898944 + 4 Combine like terms: -4.358898944 + 4 = -0.358898944 y = -0.358898944 Simplifying y = -0.358898944

Solution

The solution to the problem is based on the solutions from the subproblems. y = {8.358898944, -0.358898944}

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