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Simplifying z2 + 13z2 + -5z + 7z3 + -15z = 0 Reorder the terms: -5z + -15z + z2 + 13z2 + 7z3 = 0 Combine like terms: -5z + -15z = -20z -20z + z2 + 13z2 + 7z3 = 0 Combine like terms: z2 + 13z2 = 14z2 -20z + 14z2 + 7z3 = 0 Solving -20z + 14z2 + 7z3 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), 'z'. z(-20 + 14z + 7z2) = 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 '(-20 + 14z + 7z2)' equal to zero and attempt to solve: Simplifying -20 + 14z + 7z2 = 0 Solving -20 + 14z + 7z2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2.857142857 + 2z + z2 = 0 Move the constant term to the right: Add '2.857142857' to each side of the equation. -2.857142857 + 2z + 2.857142857 + z2 = 0 + 2.857142857 Reorder the terms: -2.857142857 + 2.857142857 + 2z + z2 = 0 + 2.857142857 Combine like terms: -2.857142857 + 2.857142857 = 0.000000000 0.000000000 + 2z + z2 = 0 + 2.857142857 2z + z2 = 0 + 2.857142857 Combine like terms: 0 + 2.857142857 = 2.857142857 2z + z2 = 2.857142857 The z term is 2z. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2z + 1 + z2 = 2.857142857 + 1 Reorder the terms: 1 + 2z + z2 = 2.857142857 + 1 Combine like terms: 2.857142857 + 1 = 3.857142857 1 + 2z + z2 = 3.857142857 Factor a perfect square on the left side: (z + 1)(z + 1) = 3.857142857 Calculate the square root of the right side: 1.963961012 Break this problem into two subproblems by setting (z + 1) equal to 1.963961012 and -1.963961012.Subproblem 1
z + 1 = 1.963961012 Simplifying z + 1 = 1.963961012 Reorder the terms: 1 + z = 1.963961012 Solving 1 + z = 1.963961012 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + z = 1.963961012 + -1 Combine like terms: 1 + -1 = 0 0 + z = 1.963961012 + -1 z = 1.963961012 + -1 Combine like terms: 1.963961012 + -1 = 0.963961012 z = 0.963961012 Simplifying z = 0.963961012Subproblem 2
z + 1 = -1.963961012 Simplifying z + 1 = -1.963961012 Reorder the terms: 1 + z = -1.963961012 Solving 1 + z = -1.963961012 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + z = -1.963961012 + -1 Combine like terms: 1 + -1 = 0 0 + z = -1.963961012 + -1 z = -1.963961012 + -1 Combine like terms: -1.963961012 + -1 = -2.963961012 z = -2.963961012 Simplifying z = -2.963961012Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.963961012, -2.963961012}Solution
z = {0, 0.963961012, -2.963961012}
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