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Simplifying 108 + -108y + -12y2 = 432 + -432y + -3y2 Solving 108 + -108y + -12y2 = 432 + -432y + -3y2 Solving for variable 'y'. Reorder the terms: 108 + -432 + -108y + 432y + -12y2 + 3y2 = 432 + -432y + -3y2 + -432 + 432y + 3y2 Combine like terms: 108 + -432 = -324 -324 + -108y + 432y + -12y2 + 3y2 = 432 + -432y + -3y2 + -432 + 432y + 3y2 Combine like terms: -108y + 432y = 324y -324 + 324y + -12y2 + 3y2 = 432 + -432y + -3y2 + -432 + 432y + 3y2 Combine like terms: -12y2 + 3y2 = -9y2 -324 + 324y + -9y2 = 432 + -432y + -3y2 + -432 + 432y + 3y2 Reorder the terms: -324 + 324y + -9y2 = 432 + -432 + -432y + 432y + -3y2 + 3y2 Combine like terms: 432 + -432 = 0 -324 + 324y + -9y2 = 0 + -432y + 432y + -3y2 + 3y2 -324 + 324y + -9y2 = -432y + 432y + -3y2 + 3y2 Combine like terms: -432y + 432y = 0 -324 + 324y + -9y2 = 0 + -3y2 + 3y2 -324 + 324y + -9y2 = -3y2 + 3y2 Combine like terms: -3y2 + 3y2 = 0 -324 + 324y + -9y2 = 0 Factor out the Greatest Common Factor (GCF), '9'. 9(-36 + 36y + -1y2) = 0 Ignore the factor 9.Subproblem 1
Set the factor '(-36 + 36y + -1y2)' equal to zero and attempt to solve: Simplifying -36 + 36y + -1y2 = 0 Solving -36 + 36y + -1y2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 36 + -36y + y2 = 0 Move the constant term to the right: Add '-36' to each side of the equation. 36 + -36y + -36 + y2 = 0 + -36 Reorder the terms: 36 + -36 + -36y + y2 = 0 + -36 Combine like terms: 36 + -36 = 0 0 + -36y + y2 = 0 + -36 -36y + y2 = 0 + -36 Combine like terms: 0 + -36 = -36 -36y + y2 = -36 The y term is -36y. Take half its coefficient (-18). Square it (324) and add it to both sides. Add '324' to each side of the equation. -36y + 324 + y2 = -36 + 324 Reorder the terms: 324 + -36y + y2 = -36 + 324 Combine like terms: -36 + 324 = 288 324 + -36y + y2 = 288 Factor a perfect square on the left side: (y + -18)(y + -18) = 288 Calculate the square root of the right side: 16.970562748 Break this problem into two subproblems by setting (y + -18) equal to 16.970562748 and -16.970562748.Subproblem 1
y + -18 = 16.970562748 Simplifying y + -18 = 16.970562748 Reorder the terms: -18 + y = 16.970562748 Solving -18 + y = 16.970562748 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '18' to each side of the equation. -18 + 18 + y = 16.970562748 + 18 Combine like terms: -18 + 18 = 0 0 + y = 16.970562748 + 18 y = 16.970562748 + 18 Combine like terms: 16.970562748 + 18 = 34.970562748 y = 34.970562748 Simplifying y = 34.970562748Subproblem 2
y + -18 = -16.970562748 Simplifying y + -18 = -16.970562748 Reorder the terms: -18 + y = -16.970562748 Solving -18 + y = -16.970562748 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '18' to each side of the equation. -18 + 18 + y = -16.970562748 + 18 Combine like terms: -18 + 18 = 0 0 + y = -16.970562748 + 18 y = -16.970562748 + 18 Combine like terms: -16.970562748 + 18 = 1.029437252 y = 1.029437252 Simplifying y = 1.029437252Solution
The solution to the problem is based on the solutions from the subproblems. y = {34.970562748, 1.029437252}Solution
y = {34.970562748, 1.029437252}
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