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Simplifying 3y2 + 10y + -51 = 0 Reorder the terms: -51 + 10y + 3y2 = 0 Solving -51 + 10y + 3y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -17 + 3.333333333y + y2 = 0 Move the constant term to the right: Add '17' to each side of the equation. -17 + 3.333333333y + 17 + y2 = 0 + 17 Reorder the terms: -17 + 17 + 3.333333333y + y2 = 0 + 17 Combine like terms: -17 + 17 = 0 0 + 3.333333333y + y2 = 0 + 17 3.333333333y + y2 = 0 + 17 Combine like terms: 0 + 17 = 17 3.333333333y + y2 = 17 The y term is 3.333333333y. Take half its coefficient (1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. 3.333333333y + 2.777777779 + y2 = 17 + 2.777777779 Reorder the terms: 2.777777779 + 3.333333333y + y2 = 17 + 2.777777779 Combine like terms: 17 + 2.777777779 = 19.777777779 2.777777779 + 3.333333333y + y2 = 19.777777779 Factor a perfect square on the left side: (y + 1.666666667)(y + 1.666666667) = 19.777777779 Calculate the square root of the right side: 4.447221355 Break this problem into two subproblems by setting (y + 1.666666667) equal to 4.447221355 and -4.447221355.Subproblem 1
y + 1.666666667 = 4.447221355 Simplifying y + 1.666666667 = 4.447221355 Reorder the terms: 1.666666667 + y = 4.447221355 Solving 1.666666667 + y = 4.447221355 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + y = 4.447221355 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + y = 4.447221355 + -1.666666667 y = 4.447221355 + -1.666666667 Combine like terms: 4.447221355 + -1.666666667 = 2.780554688 y = 2.780554688 Simplifying y = 2.780554688Subproblem 2
y + 1.666666667 = -4.447221355 Simplifying y + 1.666666667 = -4.447221355 Reorder the terms: 1.666666667 + y = -4.447221355 Solving 1.666666667 + y = -4.447221355 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + y = -4.447221355 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + y = -4.447221355 + -1.666666667 y = -4.447221355 + -1.666666667 Combine like terms: -4.447221355 + -1.666666667 = -6.113888022 y = -6.113888022 Simplifying y = -6.113888022Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.780554688, -6.113888022}
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