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Simplifying 12y2 = 40y + -3 Reorder the terms: 12y2 = -3 + 40y Solving 12y2 = -3 + 40y Solving for variable 'y'. Reorder the terms: 3 + -40y + 12y2 = -3 + 40y + 3 + -40y Reorder the terms: 3 + -40y + 12y2 = -3 + 3 + 40y + -40y Combine like terms: -3 + 3 = 0 3 + -40y + 12y2 = 0 + 40y + -40y 3 + -40y + 12y2 = 40y + -40y Combine like terms: 40y + -40y = 0 3 + -40y + 12y2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 0.25 + -3.333333333y + y2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -3.333333333y + -0.25 + y2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -3.333333333y + y2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -3.333333333y + y2 = 0 + -0.25 -3.333333333y + y2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -3.333333333y + y2 = -0.25 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 = -0.25 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333y + y2 = -0.25 + 2.777777779 Combine like terms: -0.25 + 2.777777779 = 2.527777779 2.777777779 + -3.333333333y + y2 = 2.527777779 Factor a perfect square on the left side: (y + -1.666666667)(y + -1.666666667) = 2.527777779 Calculate the square root of the right side: 1.589898669 Break this problem into two subproblems by setting (y + -1.666666667) equal to 1.589898669 and -1.589898669.Subproblem 1
y + -1.666666667 = 1.589898669 Simplifying y + -1.666666667 = 1.589898669 Reorder the terms: -1.666666667 + y = 1.589898669 Solving -1.666666667 + y = 1.589898669 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 = 1.589898669 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + y = 1.589898669 + 1.666666667 y = 1.589898669 + 1.666666667 Combine like terms: 1.589898669 + 1.666666667 = 3.256565336 y = 3.256565336 Simplifying y = 3.256565336Subproblem 2
y + -1.666666667 = -1.589898669 Simplifying y + -1.666666667 = -1.589898669 Reorder the terms: -1.666666667 + y = -1.589898669 Solving -1.666666667 + y = -1.589898669 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 = -1.589898669 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + y = -1.589898669 + 1.666666667 y = -1.589898669 + 1.666666667 Combine like terms: -1.589898669 + 1.666666667 = 0.076767998 y = 0.076767998 Simplifying y = 0.076767998Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.256565336, 0.076767998}
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