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Simplifying 8y2 + 20y + 13 = y2 + 5y + 6 Reorder the terms: 13 + 20y + 8y2 = y2 + 5y + 6 Reorder the terms: 13 + 20y + 8y2 = 6 + 5y + y2 Solving 13 + 20y + 8y2 = 6 + 5y + y2 Solving for variable 'y'. Reorder the terms: 13 + -6 + 20y + -5y + 8y2 + -1y2 = 6 + 5y + y2 + -6 + -5y + -1y2 Combine like terms: 13 + -6 = 7 7 + 20y + -5y + 8y2 + -1y2 = 6 + 5y + y2 + -6 + -5y + -1y2 Combine like terms: 20y + -5y = 15y 7 + 15y + 8y2 + -1y2 = 6 + 5y + y2 + -6 + -5y + -1y2 Combine like terms: 8y2 + -1y2 = 7y2 7 + 15y + 7y2 = 6 + 5y + y2 + -6 + -5y + -1y2 Reorder the terms: 7 + 15y + 7y2 = 6 + -6 + 5y + -5y + y2 + -1y2 Combine like terms: 6 + -6 = 0 7 + 15y + 7y2 = 0 + 5y + -5y + y2 + -1y2 7 + 15y + 7y2 = 5y + -5y + y2 + -1y2 Combine like terms: 5y + -5y = 0 7 + 15y + 7y2 = 0 + y2 + -1y2 7 + 15y + 7y2 = y2 + -1y2 Combine like terms: y2 + -1y2 = 0 7 + 15y + 7y2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 1 + 2.142857143y + y2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + 2.142857143y + -1 + y2 = 0 + -1 Reorder the terms: 1 + -1 + 2.142857143y + y2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 2.142857143y + y2 = 0 + -1 2.142857143y + y2 = 0 + -1 Combine like terms: 0 + -1 = -1 2.142857143y + y2 = -1 The y term is 2.142857143y. Take half its coefficient (1.071428572). Square it (1.147959185) and add it to both sides. Add '1.147959185' to each side of the equation. 2.142857143y + 1.147959185 + y2 = -1 + 1.147959185 Reorder the terms: 1.147959185 + 2.142857143y + y2 = -1 + 1.147959185 Combine like terms: -1 + 1.147959185 = 0.147959185 1.147959185 + 2.142857143y + y2 = 0.147959185 Factor a perfect square on the left side: (y + 1.071428572)(y + 1.071428572) = 0.147959185 Calculate the square root of the right side: 0.384654631 Break this problem into two subproblems by setting (y + 1.071428572) equal to 0.384654631 and -0.384654631.Subproblem 1
y + 1.071428572 = 0.384654631 Simplifying y + 1.071428572 = 0.384654631 Reorder the terms: 1.071428572 + y = 0.384654631 Solving 1.071428572 + y = 0.384654631 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.071428572' to each side of the equation. 1.071428572 + -1.071428572 + y = 0.384654631 + -1.071428572 Combine like terms: 1.071428572 + -1.071428572 = 0.000000000 0.000000000 + y = 0.384654631 + -1.071428572 y = 0.384654631 + -1.071428572 Combine like terms: 0.384654631 + -1.071428572 = -0.686773941 y = -0.686773941 Simplifying y = -0.686773941Subproblem 2
y + 1.071428572 = -0.384654631 Simplifying y + 1.071428572 = -0.384654631 Reorder the terms: 1.071428572 + y = -0.384654631 Solving 1.071428572 + y = -0.384654631 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.071428572' to each side of the equation. 1.071428572 + -1.071428572 + y = -0.384654631 + -1.071428572 Combine like terms: 1.071428572 + -1.071428572 = 0.000000000 0.000000000 + y = -0.384654631 + -1.071428572 y = -0.384654631 + -1.071428572 Combine like terms: -0.384654631 + -1.071428572 = -1.456083203 y = -1.456083203 Simplifying y = -1.456083203Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.686773941, -1.456083203}
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