y^2-10y=-23

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Solution for y^2-10y=-23 equation:


Simplifying
y2 + -10y = -23

Reorder the terms:
-10y + y2 = -23

Solving
-10y + y2 = -23

Solving for variable 'y'.

Reorder the terms:
23 + -10y + y2 = -23 + 23

Combine like terms: -23 + 23 = 0
23 + -10y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-23' to each side of the equation.
23 + -10y + -23 + y2 = 0 + -23

Reorder the terms:
23 + -23 + -10y + y2 = 0 + -23

Combine like terms: 23 + -23 = 0
0 + -10y + y2 = 0 + -23
-10y + y2 = 0 + -23

Combine like terms: 0 + -23 = -23
-10y + y2 = -23

The y term is -10y.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10y + 25 + y2 = -23 + 25

Reorder the terms:
25 + -10y + y2 = -23 + 25

Combine like terms: -23 + 25 = 2
25 + -10y + y2 = 2

Factor a perfect square on the left side:
(y + -5)(y + -5) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(y + -5) equal to 1.414213562 and -1.414213562.

Subproblem 1

y + -5 = 1.414213562 Simplifying y + -5 = 1.414213562 Reorder the terms: -5 + y = 1.414213562 Solving -5 + y = 1.414213562 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + y = 1.414213562 + 5 Combine like terms: -5 + 5 = 0 0 + y = 1.414213562 + 5 y = 1.414213562 + 5 Combine like terms: 1.414213562 + 5 = 6.414213562 y = 6.414213562 Simplifying y = 6.414213562

Subproblem 2

y + -5 = -1.414213562 Simplifying y + -5 = -1.414213562 Reorder the terms: -5 + y = -1.414213562 Solving -5 + y = -1.414213562 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + y = -1.414213562 + 5 Combine like terms: -5 + 5 = 0 0 + y = -1.414213562 + 5 y = -1.414213562 + 5 Combine like terms: -1.414213562 + 5 = 3.585786438 y = 3.585786438 Simplifying y = 3.585786438

Solution

The solution to the problem is based on the solutions from the subproblems. y = {6.414213562, 3.585786438}

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