10y^2+27y-15=0

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Solution for 10y^2+27y-15=0 equation:


Simplifying
10y2 + 27y + -15 = 0

Reorder the terms:
-15 + 27y + 10y2 = 0

Solving
-15 + 27y + 10y2 = 0

Solving for variable 'y'.

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

Divide each side by '10'.
-1.5 + 2.7y + y2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 2.7y + 1.5 + y2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 2.7y + y2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 2.7y + y2 = 0 + 1.5
2.7y + y2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
2.7y + y2 = 1.5

The y term is 2.7y.  Take half its coefficient (1.35).
Square it (1.8225) and add it to both sides.

Add '1.8225' to each side of the equation.
2.7y + 1.8225 + y2 = 1.5 + 1.8225

Reorder the terms:
1.8225 + 2.7y + y2 = 1.5 + 1.8225

Combine like terms: 1.5 + 1.8225 = 3.3225
1.8225 + 2.7y + y2 = 3.3225

Factor a perfect square on the left side:
(y + 1.35)(y + 1.35) = 3.3225

Calculate the square root of the right side: 1.822772613

Break this problem into two subproblems by setting 
(y + 1.35) equal to 1.822772613 and -1.822772613.

Subproblem 1

y + 1.35 = 1.822772613 Simplifying y + 1.35 = 1.822772613 Reorder the terms: 1.35 + y = 1.822772613 Solving 1.35 + y = 1.822772613 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.35' to each side of the equation. 1.35 + -1.35 + y = 1.822772613 + -1.35 Combine like terms: 1.35 + -1.35 = 0.00 0.00 + y = 1.822772613 + -1.35 y = 1.822772613 + -1.35 Combine like terms: 1.822772613 + -1.35 = 0.472772613 y = 0.472772613 Simplifying y = 0.472772613

Subproblem 2

y + 1.35 = -1.822772613 Simplifying y + 1.35 = -1.822772613 Reorder the terms: 1.35 + y = -1.822772613 Solving 1.35 + y = -1.822772613 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.35' to each side of the equation. 1.35 + -1.35 + y = -1.822772613 + -1.35 Combine like terms: 1.35 + -1.35 = 0.00 0.00 + y = -1.822772613 + -1.35 y = -1.822772613 + -1.35 Combine like terms: -1.822772613 + -1.35 = -3.172772613 y = -3.172772613 Simplifying y = -3.172772613

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.472772613, -3.172772613}

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