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Simplifying (-1z2 + -5z + 1) + (-4z2 + 3z + 1) = 0 Reorder the terms: (1 + -5z + -1z2) + (-4z2 + 3z + 1) = 0 Remove parenthesis around (1 + -5z + -1z2) 1 + -5z + -1z2 + (-4z2 + 3z + 1) = 0 Reorder the terms: 1 + -5z + -1z2 + (1 + 3z + -4z2) = 0 Remove parenthesis around (1 + 3z + -4z2) 1 + -5z + -1z2 + 1 + 3z + -4z2 = 0 Reorder the terms: 1 + 1 + -5z + 3z + -1z2 + -4z2 = 0 Combine like terms: 1 + 1 = 2 2 + -5z + 3z + -1z2 + -4z2 = 0 Combine like terms: -5z + 3z = -2z 2 + -2z + -1z2 + -4z2 = 0 Combine like terms: -1z2 + -4z2 = -5z2 2 + -2z + -5z2 = 0 Solving 2 + -2z + -5z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -0.4 + 0.4z + z2 = 0 Move the constant term to the right: Add '0.4' to each side of the equation. -0.4 + 0.4z + 0.4 + z2 = 0 + 0.4 Reorder the terms: -0.4 + 0.4 + 0.4z + z2 = 0 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + 0.4z + z2 = 0 + 0.4 0.4z + z2 = 0 + 0.4 Combine like terms: 0 + 0.4 = 0.4 0.4z + z2 = 0.4 The z term is 0.4z. Take half its coefficient (0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. 0.4z + 0.04 + z2 = 0.4 + 0.04 Reorder the terms: 0.04 + 0.4z + z2 = 0.4 + 0.04 Combine like terms: 0.4 + 0.04 = 0.44 0.04 + 0.4z + z2 = 0.44 Factor a perfect square on the left side: (z + 0.2)(z + 0.2) = 0.44 Calculate the square root of the right side: 0.663324958 Break this problem into two subproblems by setting (z + 0.2) equal to 0.663324958 and -0.663324958.Subproblem 1
z + 0.2 = 0.663324958 Simplifying z + 0.2 = 0.663324958 Reorder the terms: 0.2 + z = 0.663324958 Solving 0.2 + z = 0.663324958 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + z = 0.663324958 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + z = 0.663324958 + -0.2 z = 0.663324958 + -0.2 Combine like terms: 0.663324958 + -0.2 = 0.463324958 z = 0.463324958 Simplifying z = 0.463324958Subproblem 2
z + 0.2 = -0.663324958 Simplifying z + 0.2 = -0.663324958 Reorder the terms: 0.2 + z = -0.663324958 Solving 0.2 + z = -0.663324958 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + z = -0.663324958 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + z = -0.663324958 + -0.2 z = -0.663324958 + -0.2 Combine like terms: -0.663324958 + -0.2 = -0.863324958 z = -0.863324958 Simplifying z = -0.863324958Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.463324958, -0.863324958}
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