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Simplifying [-1(y4 + -1y2 + 1) + -1(y4 + 5y2 + 1)] + (8y4 + -6y2 + -8) = 0 Reorder the terms: [-1(1 + -1y2 + y4) + -1(y4 + 5y2 + 1)] + (8y4 + -6y2 + -8) = 0 [(1 * -1 + -1y2 * -1 + y4 * -1) + -1(y4 + 5y2 + 1)] + (8y4 + -6y2 + -8) = 0 [(-1 + 1y2 + -1y4) + -1(y4 + 5y2 + 1)] + (8y4 + -6y2 + -8) = 0 Reorder the terms: [-1 + 1y2 + -1y4 + -1(1 + 5y2 + y4)] + (8y4 + -6y2 + -8) = 0 [-1 + 1y2 + -1y4 + (1 * -1 + 5y2 * -1 + y4 * -1)] + (8y4 + -6y2 + -8) = 0 [-1 + 1y2 + -1y4 + (-1 + -5y2 + -1y4)] + (8y4 + -6y2 + -8) = 0 Reorder the terms: [-1 + -1 + 1y2 + -5y2 + -1y4 + -1y4] + (8y4 + -6y2 + -8) = 0 Combine like terms: -1 + -1 = -2 [-2 + 1y2 + -5y2 + -1y4 + -1y4] + (8y4 + -6y2 + -8) = 0 Combine like terms: 1y2 + -5y2 = -4y2 [-2 + -4y2 + -1y4 + -1y4] + (8y4 + -6y2 + -8) = 0 Combine like terms: -1y4 + -1y4 = -2y4 [-2 + -4y2 + -2y4] + (8y4 + -6y2 + -8) = 0 Remove brackets around [-2 + -4y2 + -2y4] -2 + -4y2 + -2y4 + (8y4 + -6y2 + -8) = 0 Reorder the terms: -2 + -4y2 + -2y4 + (-8 + -6y2 + 8y4) = 0 Remove parenthesis around (-8 + -6y2 + 8y4) -2 + -4y2 + -2y4 + -8 + -6y2 + 8y4 = 0 Reorder the terms: -2 + -8 + -4y2 + -6y2 + -2y4 + 8y4 = 0 Combine like terms: -2 + -8 = -10 -10 + -4y2 + -6y2 + -2y4 + 8y4 = 0 Combine like terms: -4y2 + -6y2 = -10y2 -10 + -10y2 + -2y4 + 8y4 = 0 Combine like terms: -2y4 + 8y4 = 6y4 -10 + -10y2 + 6y4 = 0 Solving -10 + -10y2 + 6y4 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '2'. 2(-5 + -5y2 + 3y4) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5 + -5y2 + 3y4)' equal to zero and attempt to solve: Simplifying -5 + -5y2 + 3y4 = 0 Solving -5 + -5y2 + 3y4 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1.666666667 + -1.666666667y2 + y4 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + -1.666666667y2 + 1.666666667 + y4 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + -1.666666667y2 + y4 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + -1.666666667y2 + y4 = 0 + 1.666666667 -1.666666667y2 + y4 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 -1.666666667y2 + y4 = 1.666666667 The y term is -1.666666667y2. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667y2 + 0.6944444447 + y4 = 1.666666667 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667y2 + y4 = 1.666666667 + 0.6944444447 Combine like terms: 1.666666667 + 0.6944444447 = 2.3611111117 0.6944444447 + -1.666666667y2 + y4 = 2.3611111117 Factor a perfect square on the left side: (y2 + -0.8333333335)(y2 + -0.8333333335) = 2.3611111117 Calculate the square root of the right side: 1.536590743 Break this problem into two subproblems by setting (y2 + -0.8333333335) equal to 1.536590743 and -1.536590743.Subproblem 1
y2 + -0.8333333335 = 1.536590743 Simplifying y2 + -0.8333333335 = 1.536590743 Reorder the terms: -0.8333333335 + y2 = 1.536590743 Solving -0.8333333335 + y2 = 1.536590743 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y2 = 1.536590743 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y2 = 1.536590743 + 0.8333333335 y2 = 1.536590743 + 0.8333333335 Combine like terms: 1.536590743 + 0.8333333335 = 2.3699240765 y2 = 2.3699240765 Simplifying y2 = 2.3699240765 Take the square root of each side: y = {-1.539455773, 1.539455773}Subproblem 2
y2 + -0.8333333335 = -1.536590743 Simplifying y2 + -0.8333333335 = -1.536590743 Reorder the terms: -0.8333333335 + y2 = -1.536590743 Solving -0.8333333335 + y2 = -1.536590743 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y2 = -1.536590743 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y2 = -1.536590743 + 0.8333333335 y2 = -1.536590743 + 0.8333333335 Combine like terms: -1.536590743 + 0.8333333335 = -0.7032574095 y2 = -0.7032574095 Simplifying y2 = -0.7032574095 Reorder the terms: 0.7032574095 + y2 = -0.7032574095 + 0.7032574095 Combine like terms: -0.7032574095 + 0.7032574095 = 0.0000000000 0.7032574095 + y2 = 0.0000000000 The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined.
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