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Simplifying 5y4 + -7y2 + 1 = 0 Reorder the terms: 1 + -7y2 + 5y4 = 0 Solving 1 + -7y2 + 5y4 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.2 + -1.4y2 + y4 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + -1.4y2 + -0.2 + y4 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + -1.4y2 + y4 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + -1.4y2 + y4 = 0 + -0.2 -1.4y2 + y4 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 -1.4y2 + y4 = -0.2 The y term is -1.4y2. Take half its coefficient (-0.7). Square it (0.49) and add it to both sides. Add '0.49' to each side of the equation. -1.4y2 + 0.49 + y4 = -0.2 + 0.49 Reorder the terms: 0.49 + -1.4y2 + y4 = -0.2 + 0.49 Combine like terms: -0.2 + 0.49 = 0.29 0.49 + -1.4y2 + y4 = 0.29 Factor a perfect square on the left side: (y2 + -0.7)(y2 + -0.7) = 0.29 Calculate the square root of the right side: 0.538516481 Break this problem into two subproblems by setting (y2 + -0.7) equal to 0.538516481 and -0.538516481.Subproblem 1
y2 + -0.7 = 0.538516481 Simplifying y2 + -0.7 = 0.538516481 Reorder the terms: -0.7 + y2 = 0.538516481 Solving -0.7 + y2 = 0.538516481 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + y2 = 0.538516481 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + y2 = 0.538516481 + 0.7 y2 = 0.538516481 + 0.7 Combine like terms: 0.538516481 + 0.7 = 1.238516481 y2 = 1.238516481 Simplifying y2 = 1.238516481 Take the square root of each side: y = {-1.112886554, 1.112886554}Subproblem 2
y2 + -0.7 = -0.538516481 Simplifying y2 + -0.7 = -0.538516481 Reorder the terms: -0.7 + y2 = -0.538516481 Solving -0.7 + y2 = -0.538516481 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + y2 = -0.538516481 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + y2 = -0.538516481 + 0.7 y2 = -0.538516481 + 0.7 Combine like terms: -0.538516481 + 0.7 = 0.161483519 y2 = 0.161483519 Simplifying y2 = 0.161483519 Take the square root of each side: y = {-0.40185012, 0.40185012}Solution
The solution to the problem is based on the solutions from the subproblems. y = {-1.112886554, 1.112886554, -0.40185012, 0.40185012}
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