10y+5-y^2=25y-10

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Solution for 10y+5-y^2=25y-10 equation:


Simplifying
10y + 5 + -1y2 = 25y + -10

Reorder the terms:
5 + 10y + -1y2 = 25y + -10

Reorder the terms:
5 + 10y + -1y2 = -10 + 25y

Solving
5 + 10y + -1y2 = -10 + 25y

Solving for variable 'y'.

Reorder the terms:
5 + 10 + 10y + -25y + -1y2 = -10 + 25y + 10 + -25y

Combine like terms: 5 + 10 = 15
15 + 10y + -25y + -1y2 = -10 + 25y + 10 + -25y

Combine like terms: 10y + -25y = -15y
15 + -15y + -1y2 = -10 + 25y + 10 + -25y

Reorder the terms:
15 + -15y + -1y2 = -10 + 10 + 25y + -25y

Combine like terms: -10 + 10 = 0
15 + -15y + -1y2 = 0 + 25y + -25y
15 + -15y + -1y2 = 25y + -25y

Combine like terms: 25y + -25y = 0
15 + -15y + -1y2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-15 + 15y + y2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 15y + 15 + y2 = 0 + 15

Reorder the terms:
-15 + 15 + 15y + y2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 15y + y2 = 0 + 15
15y + y2 = 0 + 15

Combine like terms: 0 + 15 = 15
15y + y2 = 15

The y term is 15y.  Take half its coefficient (7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
15y + 56.25 + y2 = 15 + 56.25

Reorder the terms:
56.25 + 15y + y2 = 15 + 56.25

Combine like terms: 15 + 56.25 = 71.25
56.25 + 15y + y2 = 71.25

Factor a perfect square on the left side:
(y + 7.5)(y + 7.5) = 71.25

Calculate the square root of the right side: 8.440971508

Break this problem into two subproblems by setting 
(y + 7.5) equal to 8.440971508 and -8.440971508.

Subproblem 1

y + 7.5 = 8.440971508 Simplifying y + 7.5 = 8.440971508 Reorder the terms: 7.5 + y = 8.440971508 Solving 7.5 + y = 8.440971508 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + y = 8.440971508 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + y = 8.440971508 + -7.5 y = 8.440971508 + -7.5 Combine like terms: 8.440971508 + -7.5 = 0.940971508 y = 0.940971508 Simplifying y = 0.940971508

Subproblem 2

y + 7.5 = -8.440971508 Simplifying y + 7.5 = -8.440971508 Reorder the terms: 7.5 + y = -8.440971508 Solving 7.5 + y = -8.440971508 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + y = -8.440971508 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + y = -8.440971508 + -7.5 y = -8.440971508 + -7.5 Combine like terms: -8.440971508 + -7.5 = -15.940971508 y = -15.940971508 Simplifying y = -15.940971508

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.940971508, -15.940971508}

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