3y^2=42-3y

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Solution for 3y^2=42-3y equation:


Simplifying
3y2 = 42 + -3y

Solving
3y2 = 42 + -3y

Solving for variable 'y'.

Reorder the terms:
-42 + 3y + 3y2 = 42 + -3y + -42 + 3y

Reorder the terms:
-42 + 3y + 3y2 = 42 + -42 + -3y + 3y

Combine like terms: 42 + -42 = 0
-42 + 3y + 3y2 = 0 + -3y + 3y
-42 + 3y + 3y2 = -3y + 3y

Combine like terms: -3y + 3y = 0
-42 + 3y + 3y2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-14 + y + y2) = 0

Ignore the factor 3.

Subproblem 1

Set the factor '(-14 + y + y2)' equal to zero and attempt to solve: Simplifying -14 + y + y2 = 0 Solving -14 + y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '14' to each side of the equation. -14 + y + 14 + y2 = 0 + 14 Reorder the terms: -14 + 14 + y + y2 = 0 + 14 Combine like terms: -14 + 14 = 0 0 + y + y2 = 0 + 14 y + y2 = 0 + 14 Combine like terms: 0 + 14 = 14 y + y2 = 14 The y term is y. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. y + 0.25 + y2 = 14 + 0.25 Reorder the terms: 0.25 + y + y2 = 14 + 0.25 Combine like terms: 14 + 0.25 = 14.25 0.25 + y + y2 = 14.25 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 14.25 Calculate the square root of the right side: 3.774917218 Break this problem into two subproblems by setting (y + 0.5) equal to 3.774917218 and -3.774917218.

Subproblem 1

y + 0.5 = 3.774917218 Simplifying y + 0.5 = 3.774917218 Reorder the terms: 0.5 + y = 3.774917218 Solving 0.5 + y = 3.774917218 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 3.774917218 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 3.774917218 + -0.5 y = 3.774917218 + -0.5 Combine like terms: 3.774917218 + -0.5 = 3.274917218 y = 3.274917218 Simplifying y = 3.274917218

Subproblem 2

y + 0.5 = -3.774917218 Simplifying y + 0.5 = -3.774917218 Reorder the terms: 0.5 + y = -3.774917218 Solving 0.5 + y = -3.774917218 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -3.774917218 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -3.774917218 + -0.5 y = -3.774917218 + -0.5 Combine like terms: -3.774917218 + -0.5 = -4.274917218 y = -4.274917218 Simplifying y = -4.274917218

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.274917218, -4.274917218}

Solution

y = {3.274917218, -4.274917218}

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