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Simplifying 7z(z2 + 4z + -3) + 9z(-2z2 + z) = 0 Reorder the terms: 7z(-3 + 4z + z2) + 9z(-2z2 + z) = 0 (-3 * 7z + 4z * 7z + z2 * 7z) + 9z(-2z2 + z) = 0 (-21z + 28z2 + 7z3) + 9z(-2z2 + z) = 0 Reorder the terms: -21z + 28z2 + 7z3 + 9z(z + -2z2) = 0 -21z + 28z2 + 7z3 + (z * 9z + -2z2 * 9z) = 0 -21z + 28z2 + 7z3 + (9z2 + -18z3) = 0 Reorder the terms: -21z + 28z2 + 9z2 + 7z3 + -18z3 = 0 Combine like terms: 28z2 + 9z2 = 37z2 -21z + 37z2 + 7z3 + -18z3 = 0 Combine like terms: 7z3 + -18z3 = -11z3 -21z + 37z2 + -11z3 = 0 Solving -21z + 37z2 + -11z3 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), 'z'. z(-21 + 37z + -11z2) = 0Subproblem 1
Set the factor 'z' equal to zero and attempt to solve: Simplifying z = 0 Solving z = 0 Move all terms containing z to the left, all other terms to the right. Simplifying z = 0Subproblem 2
Set the factor '(-21 + 37z + -11z2)' equal to zero and attempt to solve: Simplifying -21 + 37z + -11z2 = 0 Solving -21 + 37z + -11z2 = 0 Begin completing the square. Divide all terms by -11 the coefficient of the squared term: Divide each side by '-11'. 1.909090909 + -3.363636364z + z2 = 0 Move the constant term to the right: Add '-1.909090909' to each side of the equation. 1.909090909 + -3.363636364z + -1.909090909 + z2 = 0 + -1.909090909 Reorder the terms: 1.909090909 + -1.909090909 + -3.363636364z + z2 = 0 + -1.909090909 Combine like terms: 1.909090909 + -1.909090909 = 0.000000000 0.000000000 + -3.363636364z + z2 = 0 + -1.909090909 -3.363636364z + z2 = 0 + -1.909090909 Combine like terms: 0 + -1.909090909 = -1.909090909 -3.363636364z + z2 = -1.909090909 The z term is -3.363636364z. Take half its coefficient (-1.681818182). Square it (2.828512397) and add it to both sides. Add '2.828512397' to each side of the equation. -3.363636364z + 2.828512397 + z2 = -1.909090909 + 2.828512397 Reorder the terms: 2.828512397 + -3.363636364z + z2 = -1.909090909 + 2.828512397 Combine like terms: -1.909090909 + 2.828512397 = 0.919421488 2.828512397 + -3.363636364z + z2 = 0.919421488 Factor a perfect square on the left side: (z + -1.681818182)(z + -1.681818182) = 0.919421488 Calculate the square root of the right side: 0.958864687 Break this problem into two subproblems by setting (z + -1.681818182) equal to 0.958864687 and -0.958864687.Subproblem 1
z + -1.681818182 = 0.958864687 Simplifying z + -1.681818182 = 0.958864687 Reorder the terms: -1.681818182 + z = 0.958864687 Solving -1.681818182 + z = 0.958864687 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.681818182' to each side of the equation. -1.681818182 + 1.681818182 + z = 0.958864687 + 1.681818182 Combine like terms: -1.681818182 + 1.681818182 = 0.000000000 0.000000000 + z = 0.958864687 + 1.681818182 z = 0.958864687 + 1.681818182 Combine like terms: 0.958864687 + 1.681818182 = 2.640682869 z = 2.640682869 Simplifying z = 2.640682869Subproblem 2
z + -1.681818182 = -0.958864687 Simplifying z + -1.681818182 = -0.958864687 Reorder the terms: -1.681818182 + z = -0.958864687 Solving -1.681818182 + z = -0.958864687 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.681818182' to each side of the equation. -1.681818182 + 1.681818182 + z = -0.958864687 + 1.681818182 Combine like terms: -1.681818182 + 1.681818182 = 0.000000000 0.000000000 + z = -0.958864687 + 1.681818182 z = -0.958864687 + 1.681818182 Combine like terms: -0.958864687 + 1.681818182 = 0.722953495 z = 0.722953495 Simplifying z = 0.722953495Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.640682869, 0.722953495}Solution
z = {0, 2.640682869, 0.722953495}
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