2y^2+9y=10

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Solution for 2y^2+9y=10 equation:


Simplifying
2y2 + 9y = 10

Reorder the terms:
9y + 2y2 = 10

Solving
9y + 2y2 = 10

Solving for variable 'y'.

Reorder the terms:
-10 + 9y + 2y2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 9y + 2y2 = 0

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

Divide each side by '2'.
-5 + 4.5y + y2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 4.5y + 5 + y2 = 0 + 5

Reorder the terms:
-5 + 5 + 4.5y + y2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 4.5y + y2 = 0 + 5
4.5y + y2 = 0 + 5

Combine like terms: 0 + 5 = 5
4.5y + y2 = 5

The y term is 4.5y.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5y + 5.0625 + y2 = 5 + 5.0625

Reorder the terms:
5.0625 + 4.5y + y2 = 5 + 5.0625

Combine like terms: 5 + 5.0625 = 10.0625
5.0625 + 4.5y + y2 = 10.0625

Factor a perfect square on the left side:
(y + 2.25)(y + 2.25) = 10.0625

Calculate the square root of the right side: 3.172144385

Break this problem into two subproblems by setting 
(y + 2.25) equal to 3.172144385 and -3.172144385.

Subproblem 1

y + 2.25 = 3.172144385 Simplifying y + 2.25 = 3.172144385 Reorder the terms: 2.25 + y = 3.172144385 Solving 2.25 + y = 3.172144385 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + y = 3.172144385 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + y = 3.172144385 + -2.25 y = 3.172144385 + -2.25 Combine like terms: 3.172144385 + -2.25 = 0.922144385 y = 0.922144385 Simplifying y = 0.922144385

Subproblem 2

y + 2.25 = -3.172144385 Simplifying y + 2.25 = -3.172144385 Reorder the terms: 2.25 + y = -3.172144385 Solving 2.25 + y = -3.172144385 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + y = -3.172144385 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + y = -3.172144385 + -2.25 y = -3.172144385 + -2.25 Combine like terms: -3.172144385 + -2.25 = -5.422144385 y = -5.422144385 Simplifying y = -5.422144385

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.922144385, -5.422144385}

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