1=y+3y(y-9)

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Solution for 1=y+3y(y-9) equation:


Simplifying
1 = y + 3y(y + -9)

Reorder the terms:
1 = y + 3y(-9 + y)
1 = y + (-9 * 3y + y * 3y)
1 = y + (-27y + 3y2)

Combine like terms: y + -27y = -26y
1 = -26y + 3y2

Solving
1 = -26y + 3y2

Solving for variable 'y'.

Reorder the terms:
1 + 26y + -3y2 = -26y + 26y + 3y2 + -3y2

Combine like terms: -26y + 26y = 0
1 + 26y + -3y2 = 0 + 3y2 + -3y2
1 + 26y + -3y2 = 3y2 + -3y2

Combine like terms: 3y2 + -3y2 = 0
1 + 26y + -3y2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-0.3333333333 + -8.666666667y + y2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -8.666666667y + 0.3333333333 + y2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -8.666666667y + y2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -8.666666667y + y2 = 0 + 0.3333333333
-8.666666667y + y2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-8.666666667y + y2 = 0.3333333333

The y term is -8.666666667y.  Take half its coefficient (-4.333333334).
Square it (18.77777778) and add it to both sides.

Add '18.77777778' to each side of the equation.
-8.666666667y + 18.77777778 + y2 = 0.3333333333 + 18.77777778

Reorder the terms:
18.77777778 + -8.666666667y + y2 = 0.3333333333 + 18.77777778

Combine like terms: 0.3333333333 + 18.77777778 = 19.1111111133
18.77777778 + -8.666666667y + y2 = 19.1111111133

Factor a perfect square on the left side:
(y + -4.333333334)(y + -4.333333334) = 19.1111111133

Calculate the square root of the right side: 4.371625683

Break this problem into two subproblems by setting 
(y + -4.333333334) equal to 4.371625683 and -4.371625683.

Subproblem 1

y + -4.333333334 = 4.371625683 Simplifying y + -4.333333334 = 4.371625683 Reorder the terms: -4.333333334 + y = 4.371625683 Solving -4.333333334 + y = 4.371625683 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '4.333333334' to each side of the equation. -4.333333334 + 4.333333334 + y = 4.371625683 + 4.333333334 Combine like terms: -4.333333334 + 4.333333334 = 0.000000000 0.000000000 + y = 4.371625683 + 4.333333334 y = 4.371625683 + 4.333333334 Combine like terms: 4.371625683 + 4.333333334 = 8.704959017 y = 8.704959017 Simplifying y = 8.704959017

Subproblem 2

y + -4.333333334 = -4.371625683 Simplifying y + -4.333333334 = -4.371625683 Reorder the terms: -4.333333334 + y = -4.371625683 Solving -4.333333334 + y = -4.371625683 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '4.333333334' to each side of the equation. -4.333333334 + 4.333333334 + y = -4.371625683 + 4.333333334 Combine like terms: -4.333333334 + 4.333333334 = 0.000000000 0.000000000 + y = -4.371625683 + 4.333333334 y = -4.371625683 + 4.333333334 Combine like terms: -4.371625683 + 4.333333334 = -0.038292349 y = -0.038292349 Simplifying y = -0.038292349

Solution

The solution to the problem is based on the solutions from the subproblems. y = {8.704959017, -0.038292349}

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