10y^2-4y=45

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Solution for 10y^2-4y=45 equation:


Simplifying
10y2 + -4y = 45

Reorder the terms:
-4y + 10y2 = 45

Solving
-4y + 10y2 = 45

Solving for variable 'y'.

Reorder the terms:
-45 + -4y + 10y2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + -4y + 10y2 = 0

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

Divide each side by '10'.
-4.5 + -0.4y + y2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + -0.4y + 4.5 + y2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + -0.4y + y2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + -0.4y + y2 = 0 + 4.5
-0.4y + y2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
-0.4y + y2 = 4.5

The y term is -0.4y.  Take half its coefficient (-0.2).
Square it (0.04) and add it to both sides.

Add '0.04' to each side of the equation.
-0.4y + 0.04 + y2 = 4.5 + 0.04

Reorder the terms:
0.04 + -0.4y + y2 = 4.5 + 0.04

Combine like terms: 4.5 + 0.04 = 4.54
0.04 + -0.4y + y2 = 4.54

Factor a perfect square on the left side:
(y + -0.2)(y + -0.2) = 4.54

Calculate the square root of the right side: 2.130727575

Break this problem into two subproblems by setting 
(y + -0.2) equal to 2.130727575 and -2.130727575.

Subproblem 1

y + -0.2 = 2.130727575 Simplifying y + -0.2 = 2.130727575 Reorder the terms: -0.2 + y = 2.130727575 Solving -0.2 + y = 2.130727575 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + y = 2.130727575 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + y = 2.130727575 + 0.2 y = 2.130727575 + 0.2 Combine like terms: 2.130727575 + 0.2 = 2.330727575 y = 2.330727575 Simplifying y = 2.330727575

Subproblem 2

y + -0.2 = -2.130727575 Simplifying y + -0.2 = -2.130727575 Reorder the terms: -0.2 + y = -2.130727575 Solving -0.2 + y = -2.130727575 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + y = -2.130727575 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + y = -2.130727575 + 0.2 y = -2.130727575 + 0.2 Combine like terms: -2.130727575 + 0.2 = -1.930727575 y = -1.930727575 Simplifying y = -1.930727575

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.330727575, -1.930727575}

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