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Simplifying 9y + -4y2 = 3 Solving 9y + -4y2 = 3 Solving for variable 'y'. Reorder the terms: -3 + 9y + -4y2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 9y + -4y2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 0.75 + -2.25y + y2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + -2.25y + -0.75 + y2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + -2.25y + y2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + -2.25y + y2 = 0 + -0.75 -2.25y + y2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 -2.25y + y2 = -0.75 The y term is -2.25y. Take half its coefficient (-1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. -2.25y + 1.265625 + y2 = -0.75 + 1.265625 Reorder the terms: 1.265625 + -2.25y + y2 = -0.75 + 1.265625 Combine like terms: -0.75 + 1.265625 = 0.515625 1.265625 + -2.25y + y2 = 0.515625 Factor a perfect square on the left side: (y + -1.125)(y + -1.125) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (y + -1.125) equal to 0.718070331 and -0.718070331.Subproblem 1
y + -1.125 = 0.718070331 Simplifying y + -1.125 = 0.718070331 Reorder the terms: -1.125 + y = 0.718070331 Solving -1.125 + y = 0.718070331 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + y = 0.718070331 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + y = 0.718070331 + 1.125 y = 0.718070331 + 1.125 Combine like terms: 0.718070331 + 1.125 = 1.843070331 y = 1.843070331 Simplifying y = 1.843070331Subproblem 2
y + -1.125 = -0.718070331 Simplifying y + -1.125 = -0.718070331 Reorder the terms: -1.125 + y = -0.718070331 Solving -1.125 + y = -0.718070331 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + y = -0.718070331 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + y = -0.718070331 + 1.125 y = -0.718070331 + 1.125 Combine like terms: -0.718070331 + 1.125 = 0.406929669 y = 0.406929669 Simplifying y = 0.406929669Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.843070331, 0.406929669}
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