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Simplifying 9 = 2y(4y + -6) Reorder the terms: 9 = 2y(-6 + 4y) 9 = (-6 * 2y + 4y * 2y) 9 = (-12y + 8y2) Solving 9 = -12y + 8y2 Solving for variable 'y'. Reorder the terms: 9 + 12y + -8y2 = -12y + 12y + 8y2 + -8y2 Combine like terms: -12y + 12y = 0 9 + 12y + -8y2 = 0 + 8y2 + -8y2 9 + 12y + -8y2 = 8y2 + -8y2 Combine like terms: 8y2 + -8y2 = 0 9 + 12y + -8y2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -1.125 + -1.5y + y2 = 0 Move the constant term to the right: Add '1.125' to each side of the equation. -1.125 + -1.5y + 1.125 + y2 = 0 + 1.125 Reorder the terms: -1.125 + 1.125 + -1.5y + y2 = 0 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + -1.5y + y2 = 0 + 1.125 -1.5y + y2 = 0 + 1.125 Combine like terms: 0 + 1.125 = 1.125 -1.5y + y2 = 1.125 The y term is -1.5y. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5y + 0.5625 + y2 = 1.125 + 0.5625 Reorder the terms: 0.5625 + -1.5y + y2 = 1.125 + 0.5625 Combine like terms: 1.125 + 0.5625 = 1.6875 0.5625 + -1.5y + y2 = 1.6875 Factor a perfect square on the left side: (y + -0.75)(y + -0.75) = 1.6875 Calculate the square root of the right side: 1.299038106 Break this problem into two subproblems by setting (y + -0.75) equal to 1.299038106 and -1.299038106.Subproblem 1
y + -0.75 = 1.299038106 Simplifying y + -0.75 = 1.299038106 Reorder the terms: -0.75 + y = 1.299038106 Solving -0.75 + y = 1.299038106 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = 1.299038106 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = 1.299038106 + 0.75 y = 1.299038106 + 0.75 Combine like terms: 1.299038106 + 0.75 = 2.049038106 y = 2.049038106 Simplifying y = 2.049038106Subproblem 2
y + -0.75 = -1.299038106 Simplifying y + -0.75 = -1.299038106 Reorder the terms: -0.75 + y = -1.299038106 Solving -0.75 + y = -1.299038106 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = -1.299038106 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = -1.299038106 + 0.75 y = -1.299038106 + 0.75 Combine like terms: -1.299038106 + 0.75 = -0.549038106 y = -0.549038106 Simplifying y = -0.549038106Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.049038106, -0.549038106}
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