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Simplifying 24y2 + 13y + -78 = 0 Reorder the terms: -78 + 13y + 24y2 = 0 Solving -78 + 13y + 24y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -3.25 + 0.5416666667y + y2 = 0 Move the constant term to the right: Add '3.25' to each side of the equation. -3.25 + 0.5416666667y + 3.25 + y2 = 0 + 3.25 Reorder the terms: -3.25 + 3.25 + 0.5416666667y + y2 = 0 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + 0.5416666667y + y2 = 0 + 3.25 0.5416666667y + y2 = 0 + 3.25 Combine like terms: 0 + 3.25 = 3.25 0.5416666667y + y2 = 3.25 The y term is 0.5416666667y. Take half its coefficient (0.2708333334). Square it (0.07335069448) and add it to both sides. Add '0.07335069448' to each side of the equation. 0.5416666667y + 0.07335069448 + y2 = 3.25 + 0.07335069448 Reorder the terms: 0.07335069448 + 0.5416666667y + y2 = 3.25 + 0.07335069448 Combine like terms: 3.25 + 0.07335069448 = 3.32335069448 0.07335069448 + 0.5416666667y + y2 = 3.32335069448 Factor a perfect square on the left side: (y + 0.2708333334)(y + 0.2708333334) = 3.32335069448 Calculate the square root of the right side: 1.82300595 Break this problem into two subproblems by setting (y + 0.2708333334) equal to 1.82300595 and -1.82300595.Subproblem 1
y + 0.2708333334 = 1.82300595 Simplifying y + 0.2708333334 = 1.82300595 Reorder the terms: 0.2708333334 + y = 1.82300595 Solving 0.2708333334 + y = 1.82300595 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.2708333334' to each side of the equation. 0.2708333334 + -0.2708333334 + y = 1.82300595 + -0.2708333334 Combine like terms: 0.2708333334 + -0.2708333334 = 0.0000000000 0.0000000000 + y = 1.82300595 + -0.2708333334 y = 1.82300595 + -0.2708333334 Combine like terms: 1.82300595 + -0.2708333334 = 1.5521726166 y = 1.5521726166 Simplifying y = 1.5521726166Subproblem 2
y + 0.2708333334 = -1.82300595 Simplifying y + 0.2708333334 = -1.82300595 Reorder the terms: 0.2708333334 + y = -1.82300595 Solving 0.2708333334 + y = -1.82300595 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.2708333334' to each side of the equation. 0.2708333334 + -0.2708333334 + y = -1.82300595 + -0.2708333334 Combine like terms: 0.2708333334 + -0.2708333334 = 0.0000000000 0.0000000000 + y = -1.82300595 + -0.2708333334 y = -1.82300595 + -0.2708333334 Combine like terms: -1.82300595 + -0.2708333334 = -2.0938392834 y = -2.0938392834 Simplifying y = -2.0938392834Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.5521726166, -2.0938392834}
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