6y^2-42y+54=0

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Solution for 6y^2-42y+54=0 equation:


Simplifying
6y2 + -42y + 54 = 0

Reorder the terms:
54 + -42y + 6y2 = 0

Solving
54 + -42y + 6y2 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '6'.
6(9 + -7y + y2) = 0

Ignore the factor 6.

Subproblem 1

Set the factor '(9 + -7y + y2)' equal to zero and attempt to solve: Simplifying 9 + -7y + y2 = 0 Solving 9 + -7y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '-9' to each side of the equation. 9 + -7y + -9 + y2 = 0 + -9 Reorder the terms: 9 + -9 + -7y + y2 = 0 + -9 Combine like terms: 9 + -9 = 0 0 + -7y + y2 = 0 + -9 -7y + y2 = 0 + -9 Combine like terms: 0 + -9 = -9 -7y + y2 = -9 The y term is -7y. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7y + 12.25 + y2 = -9 + 12.25 Reorder the terms: 12.25 + -7y + y2 = -9 + 12.25 Combine like terms: -9 + 12.25 = 3.25 12.25 + -7y + y2 = 3.25 Factor a perfect square on the left side: (y + -3.5)(y + -3.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (y + -3.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

y + -3.5 = 1.802775638 Simplifying y + -3.5 = 1.802775638 Reorder the terms: -3.5 + y = 1.802775638 Solving -3.5 + y = 1.802775638 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + y = 1.802775638 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + y = 1.802775638 + 3.5 y = 1.802775638 + 3.5 Combine like terms: 1.802775638 + 3.5 = 5.302775638 y = 5.302775638 Simplifying y = 5.302775638

Subproblem 2

y + -3.5 = -1.802775638 Simplifying y + -3.5 = -1.802775638 Reorder the terms: -3.5 + y = -1.802775638 Solving -3.5 + y = -1.802775638 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + y = -1.802775638 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + y = -1.802775638 + 3.5 y = -1.802775638 + 3.5 Combine like terms: -1.802775638 + 3.5 = 1.697224362 y = 1.697224362 Simplifying y = 1.697224362

Solution

The solution to the problem is based on the solutions from the subproblems. y = {5.302775638, 1.697224362}

Solution

y = {5.302775638, 1.697224362}

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