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Simplifying z4 + -45z2 = -4 Reorder the terms: -45z2 + z4 = -4 Solving -45z2 + z4 = -4 Solving for variable 'z'. Reorder the terms: 4 + -45z2 + z4 = -4 + 4 Combine like terms: -4 + 4 = 0 4 + -45z2 + z4 = 0 Begin completing the square. Move the constant term to the right: Add '-4' to each side of the equation. 4 + -45z2 + -4 + z4 = 0 + -4 Reorder the terms: 4 + -4 + -45z2 + z4 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -45z2 + z4 = 0 + -4 -45z2 + z4 = 0 + -4 Combine like terms: 0 + -4 = -4 -45z2 + z4 = -4 The z term is -45z2. Take half its coefficient (-22.5). Square it (506.25) and add it to both sides. Add '506.25' to each side of the equation. -45z2 + 506.25 + z4 = -4 + 506.25 Reorder the terms: 506.25 + -45z2 + z4 = -4 + 506.25 Combine like terms: -4 + 506.25 = 502.25 506.25 + -45z2 + z4 = 502.25 Factor a perfect square on the left side: (z2 + -22.5)(z2 + -22.5) = 502.25 Calculate the square root of the right side: 22.410934831 Break this problem into two subproblems by setting (z2 + -22.5) equal to 22.410934831 and -22.410934831.Subproblem 1
z2 + -22.5 = 22.410934831 Simplifying z2 + -22.5 = 22.410934831 Reorder the terms: -22.5 + z2 = 22.410934831 Solving -22.5 + z2 = 22.410934831 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '22.5' to each side of the equation. -22.5 + 22.5 + z2 = 22.410934831 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + z2 = 22.410934831 + 22.5 z2 = 22.410934831 + 22.5 Combine like terms: 22.410934831 + 22.5 = 44.910934831 z2 = 44.910934831 Simplifying z2 = 44.910934831 Take the square root of each side: z = {-6.701562119, 6.701562119}Subproblem 2
z2 + -22.5 = -22.410934831 Simplifying z2 + -22.5 = -22.410934831 Reorder the terms: -22.5 + z2 = -22.410934831 Solving -22.5 + z2 = -22.410934831 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '22.5' to each side of the equation. -22.5 + 22.5 + z2 = -22.410934831 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + z2 = -22.410934831 + 22.5 z2 = -22.410934831 + 22.5 Combine like terms: -22.410934831 + 22.5 = 0.089065169 z2 = 0.089065169 Simplifying z2 = 0.089065169 Take the square root of each side: z = {-0.298437881, 0.298437881}Solution
The solution to the problem is based on the solutions from the subproblems. z = {-6.701562119, 6.701562119, -0.298437881, 0.298437881}
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