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Simplifying (-3y2 + -2y + 10) + -1(15 + -32 + 4y2) = 0 Reorder the terms: (10 + -2y + -3y2) + -1(15 + -32 + 4y2) = 0 Remove parenthesis around (10 + -2y + -3y2) 10 + -2y + -3y2 + -1(15 + -32 + 4y2) = 0 Combine like terms: 15 + -32 = -17 10 + -2y + -3y2 + -1(-17 + 4y2) = 0 10 + -2y + -3y2 + (-17 * -1 + 4y2 * -1) = 0 10 + -2y + -3y2 + (17 + -4y2) = 0 Reorder the terms: 10 + 17 + -2y + -3y2 + -4y2 = 0 Combine like terms: 10 + 17 = 27 27 + -2y + -3y2 + -4y2 = 0 Combine like terms: -3y2 + -4y2 = -7y2 27 + -2y + -7y2 = 0 Solving 27 + -2y + -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'. -3.857142857 + 0.2857142857y + y2 = 0 Move the constant term to the right: Add '3.857142857' to each side of the equation. -3.857142857 + 0.2857142857y + 3.857142857 + y2 = 0 + 3.857142857 Reorder the terms: -3.857142857 + 3.857142857 + 0.2857142857y + y2 = 0 + 3.857142857 Combine like terms: -3.857142857 + 3.857142857 = 0.000000000 0.000000000 + 0.2857142857y + y2 = 0 + 3.857142857 0.2857142857y + y2 = 0 + 3.857142857 Combine like terms: 0 + 3.857142857 = 3.857142857 0.2857142857y + y2 = 3.857142857 The y term is 0.2857142857y. Take half its coefficient (0.1428571429). Square it (0.02040816328) and add it to both sides. Add '0.02040816328' to each side of the equation. 0.2857142857y + 0.02040816328 + y2 = 3.857142857 + 0.02040816328 Reorder the terms: 0.02040816328 + 0.2857142857y + y2 = 3.857142857 + 0.02040816328 Combine like terms: 3.857142857 + 0.02040816328 = 3.87755102028 0.02040816328 + 0.2857142857y + y2 = 3.87755102028 Factor a perfect square on the left side: (y + 0.1428571429)(y + 0.1428571429) = 3.87755102028 Calculate the square root of the right side: 1.969149822 Break this problem into two subproblems by setting (y + 0.1428571429) equal to 1.969149822 and -1.969149822.Subproblem 1
y + 0.1428571429 = 1.969149822 Simplifying y + 0.1428571429 = 1.969149822 Reorder the terms: 0.1428571429 + y = 1.969149822 Solving 0.1428571429 + y = 1.969149822 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.1428571429' to each side of the equation. 0.1428571429 + -0.1428571429 + y = 1.969149822 + -0.1428571429 Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000 0.0000000000 + y = 1.969149822 + -0.1428571429 y = 1.969149822 + -0.1428571429 Combine like terms: 1.969149822 + -0.1428571429 = 1.8262926791 y = 1.8262926791 Simplifying y = 1.8262926791Subproblem 2
y + 0.1428571429 = -1.969149822 Simplifying y + 0.1428571429 = -1.969149822 Reorder the terms: 0.1428571429 + y = -1.969149822 Solving 0.1428571429 + y = -1.969149822 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.1428571429' to each side of the equation. 0.1428571429 + -0.1428571429 + y = -1.969149822 + -0.1428571429 Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000 0.0000000000 + y = -1.969149822 + -0.1428571429 y = -1.969149822 + -0.1428571429 Combine like terms: -1.969149822 + -0.1428571429 = -2.1120069649 y = -2.1120069649 Simplifying y = -2.1120069649Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.8262926791, -2.1120069649}
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