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Simplifying 9z4 + 4 = 14z2 Reorder the terms: 4 + 9z4 = 14z2 Solving 4 + 9z4 = 14z2 Solving for variable 'z'. Reorder the terms: 4 + -14z2 + 9z4 = 14z2 + -14z2 Combine like terms: 14z2 + -14z2 = 0 4 + -14z2 + 9z4 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.4444444444 + -1.555555556z2 + z4 = 0 Move the constant term to the right: Add '-0.4444444444' to each side of the equation. 0.4444444444 + -1.555555556z2 + -0.4444444444 + z4 = 0 + -0.4444444444 Reorder the terms: 0.4444444444 + -0.4444444444 + -1.555555556z2 + z4 = 0 + -0.4444444444 Combine like terms: 0.4444444444 + -0.4444444444 = 0.0000000000 0.0000000000 + -1.555555556z2 + z4 = 0 + -0.4444444444 -1.555555556z2 + z4 = 0 + -0.4444444444 Combine like terms: 0 + -0.4444444444 = -0.4444444444 -1.555555556z2 + z4 = -0.4444444444 The z term is -1.555555556z2. Take half its coefficient (-0.777777778). Square it (0.6049382720) and add it to both sides. Add '0.6049382720' to each side of the equation. -1.555555556z2 + 0.6049382720 + z4 = -0.4444444444 + 0.6049382720 Reorder the terms: 0.6049382720 + -1.555555556z2 + z4 = -0.4444444444 + 0.6049382720 Combine like terms: -0.4444444444 + 0.6049382720 = 0.1604938276 0.6049382720 + -1.555555556z2 + z4 = 0.1604938276 Factor a perfect square on the left side: (z2 + -0.777777778)(z2 + -0.777777778) = 0.1604938276 Calculate the square root of the right side: 0.400616809 Break this problem into two subproblems by setting (z2 + -0.777777778) equal to 0.400616809 and -0.400616809.Subproblem 1
z2 + -0.777777778 = 0.400616809 Simplifying z2 + -0.777777778 = 0.400616809 Reorder the terms: -0.777777778 + z2 = 0.400616809 Solving -0.777777778 + z2 = 0.400616809 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.777777778' to each side of the equation. -0.777777778 + 0.777777778 + z2 = 0.400616809 + 0.777777778 Combine like terms: -0.777777778 + 0.777777778 = 0.000000000 0.000000000 + z2 = 0.400616809 + 0.777777778 z2 = 0.400616809 + 0.777777778 Combine like terms: 0.400616809 + 0.777777778 = 1.178394587 z2 = 1.178394587 Simplifying z2 = 1.178394587 Take the square root of each side: z = {-1.085538846, 1.085538846}Subproblem 2
z2 + -0.777777778 = -0.400616809 Simplifying z2 + -0.777777778 = -0.400616809 Reorder the terms: -0.777777778 + z2 = -0.400616809 Solving -0.777777778 + z2 = -0.400616809 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.777777778' to each side of the equation. -0.777777778 + 0.777777778 + z2 = -0.400616809 + 0.777777778 Combine like terms: -0.777777778 + 0.777777778 = 0.000000000 0.000000000 + z2 = -0.400616809 + 0.777777778 z2 = -0.400616809 + 0.777777778 Combine like terms: -0.400616809 + 0.777777778 = 0.377160969 z2 = 0.377160969 Simplifying z2 = 0.377160969 Take the square root of each side: z = {-0.614134325, 0.614134325}Solution
The solution to the problem is based on the solutions from the subproblems. z = {-1.085538846, 1.085538846, -0.614134325, 0.614134325}
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