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Simplifying 11y2 + -78y + 13 = 0 Reorder the terms: 13 + -78y + 11y2 = 0 Solving 13 + -78y + 11y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. 1.181818182 + -7.090909091y + y2 = 0 Move the constant term to the right: Add '-1.181818182' to each side of the equation. 1.181818182 + -7.090909091y + -1.181818182 + y2 = 0 + -1.181818182 Reorder the terms: 1.181818182 + -1.181818182 + -7.090909091y + y2 = 0 + -1.181818182 Combine like terms: 1.181818182 + -1.181818182 = 0.000000000 0.000000000 + -7.090909091y + y2 = 0 + -1.181818182 -7.090909091y + y2 = 0 + -1.181818182 Combine like terms: 0 + -1.181818182 = -1.181818182 -7.090909091y + y2 = -1.181818182 The y term is -7.090909091y. Take half its coefficient (-3.545454546). Square it (12.57024794) and add it to both sides. Add '12.57024794' to each side of the equation. -7.090909091y + 12.57024794 + y2 = -1.181818182 + 12.57024794 Reorder the terms: 12.57024794 + -7.090909091y + y2 = -1.181818182 + 12.57024794 Combine like terms: -1.181818182 + 12.57024794 = 11.388429758 12.57024794 + -7.090909091y + y2 = 11.388429758 Factor a perfect square on the left side: (y + -3.545454546)(y + -3.545454546) = 11.388429758 Calculate the square root of the right side: 3.374674763 Break this problem into two subproblems by setting (y + -3.545454546) equal to 3.374674763 and -3.374674763.Subproblem 1
y + -3.545454546 = 3.374674763 Simplifying y + -3.545454546 = 3.374674763 Reorder the terms: -3.545454546 + y = 3.374674763 Solving -3.545454546 + y = 3.374674763 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.545454546' to each side of the equation. -3.545454546 + 3.545454546 + y = 3.374674763 + 3.545454546 Combine like terms: -3.545454546 + 3.545454546 = 0.000000000 0.000000000 + y = 3.374674763 + 3.545454546 y = 3.374674763 + 3.545454546 Combine like terms: 3.374674763 + 3.545454546 = 6.920129309 y = 6.920129309 Simplifying y = 6.920129309Subproblem 2
y + -3.545454546 = -3.374674763 Simplifying y + -3.545454546 = -3.374674763 Reorder the terms: -3.545454546 + y = -3.374674763 Solving -3.545454546 + y = -3.374674763 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.545454546' to each side of the equation. -3.545454546 + 3.545454546 + y = -3.374674763 + 3.545454546 Combine like terms: -3.545454546 + 3.545454546 = 0.000000000 0.000000000 + y = -3.374674763 + 3.545454546 y = -3.374674763 + 3.545454546 Combine like terms: -3.374674763 + 3.545454546 = 0.170779783 y = 0.170779783 Simplifying y = 0.170779783Solution
The solution to the problem is based on the solutions from the subproblems. y = {6.920129309, 0.170779783}
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