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Simplifying -2y(2y + -4) + -1y = -2(y + -2) Reorder the terms: -2y(-4 + 2y) + -1y = -2(y + -2) (-4 * -2y + 2y * -2y) + -1y = -2(y + -2) (8y + -4y2) + -1y = -2(y + -2) Reorder the terms: 8y + -1y + -4y2 = -2(y + -2) Combine like terms: 8y + -1y = 7y 7y + -4y2 = -2(y + -2) Reorder the terms: 7y + -4y2 = -2(-2 + y) 7y + -4y2 = (-2 * -2 + y * -2) 7y + -4y2 = (4 + -2y) Solving 7y + -4y2 = 4 + -2y Solving for variable 'y'. Reorder the terms: -4 + 7y + 2y + -4y2 = 4 + -2y + -4 + 2y Combine like terms: 7y + 2y = 9y -4 + 9y + -4y2 = 4 + -2y + -4 + 2y Reorder the terms: -4 + 9y + -4y2 = 4 + -4 + -2y + 2y Combine like terms: 4 + -4 = 0 -4 + 9y + -4y2 = 0 + -2y + 2y -4 + 9y + -4y2 = -2y + 2y Combine like terms: -2y + 2y = 0 -4 + 9y + -4y2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 1 + -2.25y + y2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -2.25y + -1 + y2 = 0 + -1 Reorder the terms: 1 + -1 + -2.25y + y2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2.25y + y2 = 0 + -1 -2.25y + y2 = 0 + -1 Combine like terms: 0 + -1 = -1 -2.25y + y2 = -1 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 = -1 + 1.265625 Reorder the terms: 1.265625 + -2.25y + y2 = -1 + 1.265625 Combine like terms: -1 + 1.265625 = 0.265625 1.265625 + -2.25y + y2 = 0.265625 Factor a perfect square on the left side: (y + -1.125)(y + -1.125) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (y + -1.125) equal to 0.515388203 and -0.515388203.Subproblem 1
y + -1.125 = 0.515388203 Simplifying y + -1.125 = 0.515388203 Reorder the terms: -1.125 + y = 0.515388203 Solving -1.125 + y = 0.515388203 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 = 0.515388203 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + y = 0.515388203 + 1.125 y = 0.515388203 + 1.125 Combine like terms: 0.515388203 + 1.125 = 1.640388203 y = 1.640388203 Simplifying y = 1.640388203Subproblem 2
y + -1.125 = -0.515388203 Simplifying y + -1.125 = -0.515388203 Reorder the terms: -1.125 + y = -0.515388203 Solving -1.125 + y = -0.515388203 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 = -0.515388203 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + y = -0.515388203 + 1.125 y = -0.515388203 + 1.125 Combine like terms: -0.515388203 + 1.125 = 0.609611797 y = 0.609611797 Simplifying y = 0.609611797Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.640388203, 0.609611797}
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