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Simplifying (5y2 + 3y + -6) + -1(-2y2 + -5y + 3) = 0 Reorder the terms: (-6 + 3y + 5y2) + -1(-2y2 + -5y + 3) = 0 Remove parenthesis around (-6 + 3y + 5y2) -6 + 3y + 5y2 + -1(-2y2 + -5y + 3) = 0 Reorder the terms: -6 + 3y + 5y2 + -1(3 + -5y + -2y2) = 0 -6 + 3y + 5y2 + (3 * -1 + -5y * -1 + -2y2 * -1) = 0 -6 + 3y + 5y2 + (-3 + 5y + 2y2) = 0 Reorder the terms: -6 + -3 + 3y + 5y + 5y2 + 2y2 = 0 Combine like terms: -6 + -3 = -9 -9 + 3y + 5y + 5y2 + 2y2 = 0 Combine like terms: 3y + 5y = 8y -9 + 8y + 5y2 + 2y2 = 0 Combine like terms: 5y2 + 2y2 = 7y2 -9 + 8y + 7y2 = 0 Solving -9 + 8y + 7y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.285714286 + 1.142857143y + y2 = 0 Move the constant term to the right: Add '1.285714286' to each side of the equation. -1.285714286 + 1.142857143y + 1.285714286 + y2 = 0 + 1.285714286 Reorder the terms: -1.285714286 + 1.285714286 + 1.142857143y + y2 = 0 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + 1.142857143y + y2 = 0 + 1.285714286 1.142857143y + y2 = 0 + 1.285714286 Combine like terms: 0 + 1.285714286 = 1.285714286 1.142857143y + y2 = 1.285714286 The y term is 1.142857143y. Take half its coefficient (0.5714285715). Square it (0.3265306123) and add it to both sides. Add '0.3265306123' to each side of the equation. 1.142857143y + 0.3265306123 + y2 = 1.285714286 + 0.3265306123 Reorder the terms: 0.3265306123 + 1.142857143y + y2 = 1.285714286 + 0.3265306123 Combine like terms: 1.285714286 + 0.3265306123 = 1.6122448983 0.3265306123 + 1.142857143y + y2 = 1.6122448983 Factor a perfect square on the left side: (y + 0.5714285715)(y + 0.5714285715) = 1.6122448983 Calculate the square root of the right side: 1.26974206 Break this problem into two subproblems by setting (y + 0.5714285715) equal to 1.26974206 and -1.26974206.Subproblem 1
y + 0.5714285715 = 1.26974206 Simplifying y + 0.5714285715 = 1.26974206 Reorder the terms: 0.5714285715 + y = 1.26974206 Solving 0.5714285715 + y = 1.26974206 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + y = 1.26974206 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + y = 1.26974206 + -0.5714285715 y = 1.26974206 + -0.5714285715 Combine like terms: 1.26974206 + -0.5714285715 = 0.6983134885 y = 0.6983134885 Simplifying y = 0.6983134885Subproblem 2
y + 0.5714285715 = -1.26974206 Simplifying y + 0.5714285715 = -1.26974206 Reorder the terms: 0.5714285715 + y = -1.26974206 Solving 0.5714285715 + y = -1.26974206 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + y = -1.26974206 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + y = -1.26974206 + -0.5714285715 y = -1.26974206 + -0.5714285715 Combine like terms: -1.26974206 + -0.5714285715 = -1.8411706315 y = -1.8411706315 Simplifying y = -1.8411706315Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.6983134885, -1.8411706315}
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