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Simplifying -20z2 + 20z + 5 = 0 Reorder the terms: 5 + 20z + -20z2 = 0 Solving 5 + 20z + -20z2 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '5'. 5(1 + 4z + -4z2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(1 + 4z + -4z2)' equal to zero and attempt to solve: Simplifying 1 + 4z + -4z2 = 0 Solving 1 + 4z + -4z2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.25 + -1z + z2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -1z + 0.25 + z2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -1z + z2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -1z + z2 = 0 + 0.25 -1z + z2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -1z + z2 = 0.25 The z term is -1z. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1z + 0.25 + z2 = 0.25 + 0.25 Reorder the terms: 0.25 + -1z + z2 = 0.25 + 0.25 Combine like terms: 0.25 + 0.25 = 0.5 0.25 + -1z + z2 = 0.5 Factor a perfect square on the left side: (z + -0.5)(z + -0.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (z + -0.5) equal to 0.707106781 and -0.707106781.Subproblem 1
z + -0.5 = 0.707106781 Simplifying z + -0.5 = 0.707106781 Reorder the terms: -0.5 + z = 0.707106781 Solving -0.5 + z = 0.707106781 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = 0.707106781 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = 0.707106781 + 0.5 z = 0.707106781 + 0.5 Combine like terms: 0.707106781 + 0.5 = 1.207106781 z = 1.207106781 Simplifying z = 1.207106781Subproblem 2
z + -0.5 = -0.707106781 Simplifying z + -0.5 = -0.707106781 Reorder the terms: -0.5 + z = -0.707106781 Solving -0.5 + z = -0.707106781 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = -0.707106781 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = -0.707106781 + 0.5 z = -0.707106781 + 0.5 Combine like terms: -0.707106781 + 0.5 = -0.207106781 z = -0.207106781 Simplifying z = -0.207106781Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.207106781, -0.207106781}Solution
z = {1.207106781, -0.207106781}
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