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Simplifying 0 = 9y2 + -66y + 97 Reorder the terms: 0 = 97 + -66y + 9y2 Solving 0 = 97 + -66y + 9y2 Solving for variable 'y'. Combine like terms: 0 + -97 = -97 -97 + 66y + -9y2 = 97 + -66y + 9y2 + -97 + 66y + -9y2 Reorder the terms: -97 + 66y + -9y2 = 97 + -97 + -66y + 66y + 9y2 + -9y2 Combine like terms: 97 + -97 = 0 -97 + 66y + -9y2 = 0 + -66y + 66y + 9y2 + -9y2 -97 + 66y + -9y2 = -66y + 66y + 9y2 + -9y2 Combine like terms: -66y + 66y = 0 -97 + 66y + -9y2 = 0 + 9y2 + -9y2 -97 + 66y + -9y2 = 9y2 + -9y2 Combine like terms: 9y2 + -9y2 = 0 -97 + 66y + -9y2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. 10.77777778 + -7.333333333y + y2 = 0 Move the constant term to the right: Add '-10.77777778' to each side of the equation. 10.77777778 + -7.333333333y + -10.77777778 + y2 = 0 + -10.77777778 Reorder the terms: 10.77777778 + -10.77777778 + -7.333333333y + y2 = 0 + -10.77777778 Combine like terms: 10.77777778 + -10.77777778 = 0.00000000 0.00000000 + -7.333333333y + y2 = 0 + -10.77777778 -7.333333333y + y2 = 0 + -10.77777778 Combine like terms: 0 + -10.77777778 = -10.77777778 -7.333333333y + y2 = -10.77777778 The y term is -7.333333333y. Take half its coefficient (-3.666666667). Square it (13.44444445) and add it to both sides. Add '13.44444445' to each side of the equation. -7.333333333y + 13.44444445 + y2 = -10.77777778 + 13.44444445 Reorder the terms: 13.44444445 + -7.333333333y + y2 = -10.77777778 + 13.44444445 Combine like terms: -10.77777778 + 13.44444445 = 2.66666667 13.44444445 + -7.333333333y + y2 = 2.66666667 Factor a perfect square on the left side: (y + -3.666666667)(y + -3.666666667) = 2.66666667 Calculate the square root of the right side: 1.632993163 Break this problem into two subproblems by setting (y + -3.666666667) equal to 1.632993163 and -1.632993163.Subproblem 1
y + -3.666666667 = 1.632993163 Simplifying y + -3.666666667 = 1.632993163 Reorder the terms: -3.666666667 + y = 1.632993163 Solving -3.666666667 + y = 1.632993163 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.666666667' to each side of the equation. -3.666666667 + 3.666666667 + y = 1.632993163 + 3.666666667 Combine like terms: -3.666666667 + 3.666666667 = 0.000000000 0.000000000 + y = 1.632993163 + 3.666666667 y = 1.632993163 + 3.666666667 Combine like terms: 1.632993163 + 3.666666667 = 5.29965983 y = 5.29965983 Simplifying y = 5.29965983Subproblem 2
y + -3.666666667 = -1.632993163 Simplifying y + -3.666666667 = -1.632993163 Reorder the terms: -3.666666667 + y = -1.632993163 Solving -3.666666667 + y = -1.632993163 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.666666667' to each side of the equation. -3.666666667 + 3.666666667 + y = -1.632993163 + 3.666666667 Combine like terms: -3.666666667 + 3.666666667 = 0.000000000 0.000000000 + y = -1.632993163 + 3.666666667 y = -1.632993163 + 3.666666667 Combine like terms: -1.632993163 + 3.666666667 = 2.033673504 y = 2.033673504 Simplifying y = 2.033673504Solution
The solution to the problem is based on the solutions from the subproblems. y = {5.29965983, 2.033673504}
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