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Simplifying 3z4 + -15z3 + -9z2 = 0 Reorder the terms: -9z2 + -15z3 + 3z4 = 0 Solving -9z2 + -15z3 + 3z4 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '3z2'. 3z2(-3 + -5z + z2) = 0 Ignore the factor 3.Subproblem 1
Set the factor 'z2' equal to zero and attempt to solve: Simplifying z2 = 0 Solving z2 = 0 Move all terms containing z to the left, all other terms to the right. Simplifying z2 = 0 Take the square root of each side: z = {0}Subproblem 2
Set the factor '(-3 + -5z + z2)' equal to zero and attempt to solve: Simplifying -3 + -5z + z2 = 0 Solving -3 + -5z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + -5z + 3 + z2 = 0 + 3 Reorder the terms: -3 + 3 + -5z + z2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -5z + z2 = 0 + 3 -5z + z2 = 0 + 3 Combine like terms: 0 + 3 = 3 -5z + z2 = 3 The z term is -5z. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5z + 6.25 + z2 = 3 + 6.25 Reorder the terms: 6.25 + -5z + z2 = 3 + 6.25 Combine like terms: 3 + 6.25 = 9.25 6.25 + -5z + z2 = 9.25 Factor a perfect square on the left side: (z + -2.5)(z + -2.5) = 9.25 Calculate the square root of the right side: 3.041381265 Break this problem into two subproblems by setting (z + -2.5) equal to 3.041381265 and -3.041381265.Subproblem 1
z + -2.5 = 3.041381265 Simplifying z + -2.5 = 3.041381265 Reorder the terms: -2.5 + z = 3.041381265 Solving -2.5 + z = 3.041381265 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = 3.041381265 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = 3.041381265 + 2.5 z = 3.041381265 + 2.5 Combine like terms: 3.041381265 + 2.5 = 5.541381265 z = 5.541381265 Simplifying z = 5.541381265Subproblem 2
z + -2.5 = -3.041381265 Simplifying z + -2.5 = -3.041381265 Reorder the terms: -2.5 + z = -3.041381265 Solving -2.5 + z = -3.041381265 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = -3.041381265 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = -3.041381265 + 2.5 z = -3.041381265 + 2.5 Combine like terms: -3.041381265 + 2.5 = -0.541381265 z = -0.541381265 Simplifying z = -0.541381265Solution
The solution to the problem is based on the solutions from the subproblems. z = {5.541381265, -0.541381265}Solution
z = {0, 5.541381265, -0.541381265}
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