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Simplifying 8y2 + -7y + 12 + -15y2 + -7y + 3 = 0 Reorder the terms: 12 + 3 + -7y + -7y + 8y2 + -15y2 = 0 Combine like terms: 12 + 3 = 15 15 + -7y + -7y + 8y2 + -15y2 = 0 Combine like terms: -7y + -7y = -14y 15 + -14y + 8y2 + -15y2 = 0 Combine like terms: 8y2 + -15y2 = -7y2 15 + -14y + -7y2 = 0 Solving 15 + -14y + -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'. -2.142857143 + 2y + y2 = 0 Move the constant term to the right: Add '2.142857143' to each side of the equation. -2.142857143 + 2y + 2.142857143 + y2 = 0 + 2.142857143 Reorder the terms: -2.142857143 + 2.142857143 + 2y + y2 = 0 + 2.142857143 Combine like terms: -2.142857143 + 2.142857143 = 0.000000000 0.000000000 + 2y + y2 = 0 + 2.142857143 2y + y2 = 0 + 2.142857143 Combine like terms: 0 + 2.142857143 = 2.142857143 2y + y2 = 2.142857143 The y term is 2y. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2y + 1 + y2 = 2.142857143 + 1 Reorder the terms: 1 + 2y + y2 = 2.142857143 + 1 Combine like terms: 2.142857143 + 1 = 3.142857143 1 + 2y + y2 = 3.142857143 Factor a perfect square on the left side: (y + 1)(y + 1) = 3.142857143 Calculate the square root of the right side: 1.772810521 Break this problem into two subproblems by setting (y + 1) equal to 1.772810521 and -1.772810521.Subproblem 1
y + 1 = 1.772810521 Simplifying y + 1 = 1.772810521 Reorder the terms: 1 + y = 1.772810521 Solving 1 + y = 1.772810521 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = 1.772810521 + -1 Combine like terms: 1 + -1 = 0 0 + y = 1.772810521 + -1 y = 1.772810521 + -1 Combine like terms: 1.772810521 + -1 = 0.772810521 y = 0.772810521 Simplifying y = 0.772810521Subproblem 2
y + 1 = -1.772810521 Simplifying y + 1 = -1.772810521 Reorder the terms: 1 + y = -1.772810521 Solving 1 + y = -1.772810521 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = -1.772810521 + -1 Combine like terms: 1 + -1 = 0 0 + y = -1.772810521 + -1 y = -1.772810521 + -1 Combine like terms: -1.772810521 + -1 = -2.772810521 y = -2.772810521 Simplifying y = -2.772810521Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.772810521, -2.772810521}
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