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Simplifying 8y4 + 5 + -16y2 = 0 Reorder the terms: 5 + -16y2 + 8y4 = 0 Solving 5 + -16y2 + 8y4 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.625 + -2y2 + y4 = 0 Move the constant term to the right: Add '-0.625' to each side of the equation. 0.625 + -2y2 + -0.625 + y4 = 0 + -0.625 Reorder the terms: 0.625 + -0.625 + -2y2 + y4 = 0 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + -2y2 + y4 = 0 + -0.625 -2y2 + y4 = 0 + -0.625 Combine like terms: 0 + -0.625 = -0.625 -2y2 + y4 = -0.625 The y term is -2y2. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2y2 + 1 + y4 = -0.625 + 1 Reorder the terms: 1 + -2y2 + y4 = -0.625 + 1 Combine like terms: -0.625 + 1 = 0.375 1 + -2y2 + y4 = 0.375 Factor a perfect square on the left side: (y2 + -1)(y2 + -1) = 0.375 Calculate the square root of the right side: 0.612372436 Break this problem into two subproblems by setting (y2 + -1) equal to 0.612372436 and -0.612372436.Subproblem 1
y2 + -1 = 0.612372436 Simplifying y2 + -1 = 0.612372436 Reorder the terms: -1 + y2 = 0.612372436 Solving -1 + y2 = 0.612372436 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 + y2 = 0.612372436 + 1 Combine like terms: -1 + 1 = 0 0 + y2 = 0.612372436 + 1 y2 = 0.612372436 + 1 Combine like terms: 0.612372436 + 1 = 1.612372436 y2 = 1.612372436 Simplifying y2 = 1.612372436 Take the square root of each side: y = {-1.269792281, 1.269792281}Subproblem 2
y2 + -1 = -0.612372436 Simplifying y2 + -1 = -0.612372436 Reorder the terms: -1 + y2 = -0.612372436 Solving -1 + y2 = -0.612372436 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 + y2 = -0.612372436 + 1 Combine like terms: -1 + 1 = 0 0 + y2 = -0.612372436 + 1 y2 = -0.612372436 + 1 Combine like terms: -0.612372436 + 1 = 0.387627564 y2 = 0.387627564 Simplifying y2 = 0.387627564 Take the square root of each side: y = {-0.622597433, 0.622597433}Solution
The solution to the problem is based on the solutions from the subproblems. y = {-1.269792281, 1.269792281, -0.622597433, 0.622597433}
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