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Simplifying 4y2 + -7y + 13 + -15y2 + -7y + 6 = 0 Reorder the terms: 13 + 6 + -7y + -7y + 4y2 + -15y2 = 0 Combine like terms: 13 + 6 = 19 19 + -7y + -7y + 4y2 + -15y2 = 0 Combine like terms: -7y + -7y = -14y 19 + -14y + 4y2 + -15y2 = 0 Combine like terms: 4y2 + -15y2 = -11y2 19 + -14y + -11y2 = 0 Solving 19 + -14y + -11y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by -11 the coefficient of the squared term: Divide each side by '-11'. -1.727272727 + 1.272727273y + y2 = 0 Move the constant term to the right: Add '1.727272727' to each side of the equation. -1.727272727 + 1.272727273y + 1.727272727 + y2 = 0 + 1.727272727 Reorder the terms: -1.727272727 + 1.727272727 + 1.272727273y + y2 = 0 + 1.727272727 Combine like terms: -1.727272727 + 1.727272727 = 0.000000000 0.000000000 + 1.272727273y + y2 = 0 + 1.727272727 1.272727273y + y2 = 0 + 1.727272727 Combine like terms: 0 + 1.727272727 = 1.727272727 1.272727273y + y2 = 1.727272727 The y term is 1.272727273y. Take half its coefficient (0.6363636365). Square it (0.4049586779) and add it to both sides. Add '0.4049586779' to each side of the equation. 1.272727273y + 0.4049586779 + y2 = 1.727272727 + 0.4049586779 Reorder the terms: 0.4049586779 + 1.272727273y + y2 = 1.727272727 + 0.4049586779 Combine like terms: 1.727272727 + 0.4049586779 = 2.1322314049 0.4049586779 + 1.272727273y + y2 = 2.1322314049 Factor a perfect square on the left side: (y + 0.6363636365)(y + 0.6363636365) = 2.1322314049 Calculate the square root of the right side: 1.460216219 Break this problem into two subproblems by setting (y + 0.6363636365) equal to 1.460216219 and -1.460216219.Subproblem 1
y + 0.6363636365 = 1.460216219 Simplifying y + 0.6363636365 = 1.460216219 Reorder the terms: 0.6363636365 + y = 1.460216219 Solving 0.6363636365 + y = 1.460216219 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6363636365' to each side of the equation. 0.6363636365 + -0.6363636365 + y = 1.460216219 + -0.6363636365 Combine like terms: 0.6363636365 + -0.6363636365 = 0.0000000000 0.0000000000 + y = 1.460216219 + -0.6363636365 y = 1.460216219 + -0.6363636365 Combine like terms: 1.460216219 + -0.6363636365 = 0.8238525825 y = 0.8238525825 Simplifying y = 0.8238525825Subproblem 2
y + 0.6363636365 = -1.460216219 Simplifying y + 0.6363636365 = -1.460216219 Reorder the terms: 0.6363636365 + y = -1.460216219 Solving 0.6363636365 + y = -1.460216219 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6363636365' to each side of the equation. 0.6363636365 + -0.6363636365 + y = -1.460216219 + -0.6363636365 Combine like terms: 0.6363636365 + -0.6363636365 = 0.0000000000 0.0000000000 + y = -1.460216219 + -0.6363636365 y = -1.460216219 + -0.6363636365 Combine like terms: -1.460216219 + -0.6363636365 = -2.0965798555 y = -2.0965798555 Simplifying y = -2.0965798555Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.8238525825, -2.0965798555}
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