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Simplifying 8y2 + -7y + 12 + -12y2 + -7y + 7 = 0 Reorder the terms: 12 + 7 + -7y + -7y + 8y2 + -12y2 = 0 Combine like terms: 12 + 7 = 19 19 + -7y + -7y + 8y2 + -12y2 = 0 Combine like terms: -7y + -7y = -14y 19 + -14y + 8y2 + -12y2 = 0 Combine like terms: 8y2 + -12y2 = -4y2 19 + -14y + -4y2 = 0 Solving 19 + -14y + -4y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -4.75 + 3.5y + y2 = 0 Move the constant term to the right: Add '4.75' to each side of the equation. -4.75 + 3.5y + 4.75 + y2 = 0 + 4.75 Reorder the terms: -4.75 + 4.75 + 3.5y + y2 = 0 + 4.75 Combine like terms: -4.75 + 4.75 = 0.00 0.00 + 3.5y + y2 = 0 + 4.75 3.5y + y2 = 0 + 4.75 Combine like terms: 0 + 4.75 = 4.75 3.5y + y2 = 4.75 The y term is 3.5y. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5y + 3.0625 + y2 = 4.75 + 3.0625 Reorder the terms: 3.0625 + 3.5y + y2 = 4.75 + 3.0625 Combine like terms: 4.75 + 3.0625 = 7.8125 3.0625 + 3.5y + y2 = 7.8125 Factor a perfect square on the left side: (y + 1.75)(y + 1.75) = 7.8125 Calculate the square root of the right side: 2.795084972 Break this problem into two subproblems by setting (y + 1.75) equal to 2.795084972 and -2.795084972.Subproblem 1
y + 1.75 = 2.795084972 Simplifying y + 1.75 = 2.795084972 Reorder the terms: 1.75 + y = 2.795084972 Solving 1.75 + y = 2.795084972 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + y = 2.795084972 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + y = 2.795084972 + -1.75 y = 2.795084972 + -1.75 Combine like terms: 2.795084972 + -1.75 = 1.045084972 y = 1.045084972 Simplifying y = 1.045084972Subproblem 2
y + 1.75 = -2.795084972 Simplifying y + 1.75 = -2.795084972 Reorder the terms: 1.75 + y = -2.795084972 Solving 1.75 + y = -2.795084972 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + y = -2.795084972 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + y = -2.795084972 + -1.75 y = -2.795084972 + -1.75 Combine like terms: -2.795084972 + -1.75 = -4.545084972 y = -4.545084972 Simplifying y = -4.545084972Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.045084972, -4.545084972}
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