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Simplifying 54y + -6y2 = 72 Solving 54y + -6y2 = 72 Solving for variable 'y'. Reorder the terms: -72 + 54y + -6y2 = 72 + -72 Combine like terms: 72 + -72 = 0 -72 + 54y + -6y2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-12 + 9y + -1y2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-12 + 9y + -1y2)' equal to zero and attempt to solve: Simplifying -12 + 9y + -1y2 = 0 Solving -12 + 9y + -1y2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 12 + -9y + y2 = 0 Move the constant term to the right: Add '-12' to each side of the equation. 12 + -9y + -12 + y2 = 0 + -12 Reorder the terms: 12 + -12 + -9y + y2 = 0 + -12 Combine like terms: 12 + -12 = 0 0 + -9y + y2 = 0 + -12 -9y + y2 = 0 + -12 Combine like terms: 0 + -12 = -12 -9y + y2 = -12 The y term is -9y. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9y + 20.25 + y2 = -12 + 20.25 Reorder the terms: 20.25 + -9y + y2 = -12 + 20.25 Combine like terms: -12 + 20.25 = 8.25 20.25 + -9y + y2 = 8.25 Factor a perfect square on the left side: (y + -4.5)(y + -4.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (y + -4.5) equal to 2.872281323 and -2.872281323.Subproblem 1
y + -4.5 = 2.872281323 Simplifying y + -4.5 = 2.872281323 Reorder the terms: -4.5 + y = 2.872281323 Solving -4.5 + y = 2.872281323 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + y = 2.872281323 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + y = 2.872281323 + 4.5 y = 2.872281323 + 4.5 Combine like terms: 2.872281323 + 4.5 = 7.372281323 y = 7.372281323 Simplifying y = 7.372281323Subproblem 2
y + -4.5 = -2.872281323 Simplifying y + -4.5 = -2.872281323 Reorder the terms: -4.5 + y = -2.872281323 Solving -4.5 + y = -2.872281323 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + y = -2.872281323 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + y = -2.872281323 + 4.5 y = -2.872281323 + 4.5 Combine like terms: -2.872281323 + 4.5 = 1.627718677 y = 1.627718677 Simplifying y = 1.627718677Solution
The solution to the problem is based on the solutions from the subproblems. y = {7.372281323, 1.627718677}Solution
y = {7.372281323, 1.627718677}
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