6y(3y+9)=17

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Solution for 6y(3y+9)=17 equation:


Simplifying
6y(3y + 9) = 17

Reorder the terms:
6y(9 + 3y) = 17
(9 * 6y + 3y * 6y) = 17
(54y + 18y2) = 17

Solving
54y + 18y2 = 17

Solving for variable 'y'.

Reorder the terms:
-17 + 54y + 18y2 = 17 + -17

Combine like terms: 17 + -17 = 0
-17 + 54y + 18y2 = 0

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

Divide each side by '18'.
-0.9444444444 + 3y + y2 = 0

Move the constant term to the right:

Add '0.9444444444' to each side of the equation.
-0.9444444444 + 3y + 0.9444444444 + y2 = 0 + 0.9444444444

Reorder the terms:
-0.9444444444 + 0.9444444444 + 3y + y2 = 0 + 0.9444444444

Combine like terms: -0.9444444444 + 0.9444444444 = 0.0000000000
0.0000000000 + 3y + y2 = 0 + 0.9444444444
3y + y2 = 0 + 0.9444444444

Combine like terms: 0 + 0.9444444444 = 0.9444444444
3y + y2 = 0.9444444444

The y term is 3y.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3y + 2.25 + y2 = 0.9444444444 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 0.9444444444 + 2.25

Combine like terms: 0.9444444444 + 2.25 = 3.1944444444
2.25 + 3y + y2 = 3.1944444444

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 3.1944444444

Calculate the square root of the right side: 1.787300882

Break this problem into two subproblems by setting 
(y + 1.5) equal to 1.787300882 and -1.787300882.

Subproblem 1

y + 1.5 = 1.787300882 Simplifying y + 1.5 = 1.787300882 Reorder the terms: 1.5 + y = 1.787300882 Solving 1.5 + y = 1.787300882 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = 1.787300882 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 1.787300882 + -1.5 y = 1.787300882 + -1.5 Combine like terms: 1.787300882 + -1.5 = 0.287300882 y = 0.287300882 Simplifying y = 0.287300882

Subproblem 2

y + 1.5 = -1.787300882 Simplifying y + 1.5 = -1.787300882 Reorder the terms: 1.5 + y = -1.787300882 Solving 1.5 + y = -1.787300882 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = -1.787300882 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -1.787300882 + -1.5 y = -1.787300882 + -1.5 Combine like terms: -1.787300882 + -1.5 = -3.287300882 y = -3.287300882 Simplifying y = -3.287300882

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.287300882, -3.287300882}

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