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Simplifying 2z2 + 22z = -10 Reorder the terms: 22z + 2z2 = -10 Solving 22z + 2z2 = -10 Solving for variable 'z'. Reorder the terms: 10 + 22z + 2z2 = -10 + 10 Combine like terms: -10 + 10 = 0 10 + 22z + 2z2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(5 + 11z + z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(5 + 11z + z2)' equal to zero and attempt to solve: Simplifying 5 + 11z + z2 = 0 Solving 5 + 11z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + 11z + -5 + z2 = 0 + -5 Reorder the terms: 5 + -5 + 11z + z2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 11z + z2 = 0 + -5 11z + z2 = 0 + -5 Combine like terms: 0 + -5 = -5 11z + z2 = -5 The z term is 11z. Take half its coefficient (5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. 11z + 30.25 + z2 = -5 + 30.25 Reorder the terms: 30.25 + 11z + z2 = -5 + 30.25 Combine like terms: -5 + 30.25 = 25.25 30.25 + 11z + z2 = 25.25 Factor a perfect square on the left side: (z + 5.5)(z + 5.5) = 25.25 Calculate the square root of the right side: 5.024937811 Break this problem into two subproblems by setting (z + 5.5) equal to 5.024937811 and -5.024937811.Subproblem 1
z + 5.5 = 5.024937811 Simplifying z + 5.5 = 5.024937811 Reorder the terms: 5.5 + z = 5.024937811 Solving 5.5 + z = 5.024937811 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + z = 5.024937811 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + z = 5.024937811 + -5.5 z = 5.024937811 + -5.5 Combine like terms: 5.024937811 + -5.5 = -0.475062189 z = -0.475062189 Simplifying z = -0.475062189Subproblem 2
z + 5.5 = -5.024937811 Simplifying z + 5.5 = -5.024937811 Reorder the terms: 5.5 + z = -5.024937811 Solving 5.5 + z = -5.024937811 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + z = -5.024937811 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + z = -5.024937811 + -5.5 z = -5.024937811 + -5.5 Combine like terms: -5.024937811 + -5.5 = -10.524937811 z = -10.524937811 Simplifying z = -10.524937811Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.475062189, -10.524937811}Solution
z = {-0.475062189, -10.524937811}
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