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Simplifying 3y2 + 8y + -78 = 0 Reorder the terms: -78 + 8y + 3y2 = 0 Solving -78 + 8y + 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'. -26 + 2.666666667y + y2 = 0 Move the constant term to the right: Add '26' to each side of the equation. -26 + 2.666666667y + 26 + y2 = 0 + 26 Reorder the terms: -26 + 26 + 2.666666667y + y2 = 0 + 26 Combine like terms: -26 + 26 = 0 0 + 2.666666667y + y2 = 0 + 26 2.666666667y + y2 = 0 + 26 Combine like terms: 0 + 26 = 26 2.666666667y + y2 = 26 The y term is 2.666666667y. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667y + 1.777777780 + y2 = 26 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667y + y2 = 26 + 1.777777780 Combine like terms: 26 + 1.777777780 = 27.77777778 1.777777780 + 2.666666667y + y2 = 27.77777778 Factor a perfect square on the left side: (y + 1.333333334)(y + 1.333333334) = 27.77777778 Calculate the square root of the right side: 5.270462767 Break this problem into two subproblems by setting (y + 1.333333334) equal to 5.270462767 and -5.270462767.Subproblem 1
y + 1.333333334 = 5.270462767 Simplifying y + 1.333333334 = 5.270462767 Reorder the terms: 1.333333334 + y = 5.270462767 Solving 1.333333334 + y = 5.270462767 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + y = 5.270462767 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + y = 5.270462767 + -1.333333334 y = 5.270462767 + -1.333333334 Combine like terms: 5.270462767 + -1.333333334 = 3.937129433 y = 3.937129433 Simplifying y = 3.937129433Subproblem 2
y + 1.333333334 = -5.270462767 Simplifying y + 1.333333334 = -5.270462767 Reorder the terms: 1.333333334 + y = -5.270462767 Solving 1.333333334 + y = -5.270462767 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + y = -5.270462767 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + y = -5.270462767 + -1.333333334 y = -5.270462767 + -1.333333334 Combine like terms: -5.270462767 + -1.333333334 = -6.603796101 y = -6.603796101 Simplifying y = -6.603796101Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.937129433, -6.603796101}
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