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Simplifying 6z2(7z2 + 3z + -2) = 0 Reorder the terms: 6z2(-2 + 3z + 7z2) = 0 (-2 * 6z2 + 3z * 6z2 + 7z2 * 6z2) = 0 (-12z2 + 18z3 + 42z4) = 0 Solving -12z2 + 18z3 + 42z4 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '6z2'. 6z2(-2 + 3z + 7z2) = 0 Ignore the factor 6.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 '(-2 + 3z + 7z2)' equal to zero and attempt to solve: Simplifying -2 + 3z + 7z2 = 0 Solving -2 + 3z + 7z2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.2857142857 + 0.4285714286z + z2 = 0 Move the constant term to the right: Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.4285714286z + 0.2857142857 + z2 = 0 + 0.2857142857 Reorder the terms: -0.2857142857 + 0.2857142857 + 0.4285714286z + z2 = 0 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + 0.4285714286z + z2 = 0 + 0.2857142857 0.4285714286z + z2 = 0 + 0.2857142857 Combine like terms: 0 + 0.2857142857 = 0.2857142857 0.4285714286z + z2 = 0.2857142857 The z term is 0.4285714286z. Take half its coefficient (0.2142857143). Square it (0.04591836735) and add it to both sides. Add '0.04591836735' to each side of the equation. 0.4285714286z + 0.04591836735 + z2 = 0.2857142857 + 0.04591836735 Reorder the terms: 0.04591836735 + 0.4285714286z + z2 = 0.2857142857 + 0.04591836735 Combine like terms: 0.2857142857 + 0.04591836735 = 0.33163265305 0.04591836735 + 0.4285714286z + z2 = 0.33163265305 Factor a perfect square on the left side: (z + 0.2142857143)(z + 0.2142857143) = 0.33163265305 Calculate the square root of the right side: 0.575875553 Break this problem into two subproblems by setting (z + 0.2142857143) equal to 0.575875553 and -0.575875553.Subproblem 1
z + 0.2142857143 = 0.575875553 Simplifying z + 0.2142857143 = 0.575875553 Reorder the terms: 0.2142857143 + z = 0.575875553 Solving 0.2142857143 + z = 0.575875553 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + z = 0.575875553 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + z = 0.575875553 + -0.2142857143 z = 0.575875553 + -0.2142857143 Combine like terms: 0.575875553 + -0.2142857143 = 0.3615898387 z = 0.3615898387 Simplifying z = 0.3615898387Subproblem 2
z + 0.2142857143 = -0.575875553 Simplifying z + 0.2142857143 = -0.575875553 Reorder the terms: 0.2142857143 + z = -0.575875553 Solving 0.2142857143 + z = -0.575875553 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + z = -0.575875553 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + z = -0.575875553 + -0.2142857143 z = -0.575875553 + -0.2142857143 Combine like terms: -0.575875553 + -0.2142857143 = -0.7901612673 z = -0.7901612673 Simplifying z = -0.7901612673Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.3615898387, -0.7901612673}Solution
z = {0, 0.3615898387, -0.7901612673}
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