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Simplifying 7z(z + -4) + 4z + -2 = 0 Reorder the terms: 7z(-4 + z) + 4z + -2 = 0 (-4 * 7z + z * 7z) + 4z + -2 = 0 (-28z + 7z2) + 4z + -2 = 0 Reorder the terms: -2 + -28z + 4z + 7z2 = 0 Combine like terms: -28z + 4z = -24z -2 + -24z + 7z2 = 0 Solving -2 + -24z + 7z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.2857142857 + -3.428571429z + z2 = 0 Move the constant term to the right: Add '0.2857142857' to each side of the equation. -0.2857142857 + -3.428571429z + 0.2857142857 + z2 = 0 + 0.2857142857 Reorder the terms: -0.2857142857 + 0.2857142857 + -3.428571429z + z2 = 0 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + -3.428571429z + z2 = 0 + 0.2857142857 -3.428571429z + z2 = 0 + 0.2857142857 Combine like terms: 0 + 0.2857142857 = 0.2857142857 -3.428571429z + z2 = 0.2857142857 The z term is -3.428571429z. Take half its coefficient (-1.714285715). Square it (2.938775513) and add it to both sides. Add '2.938775513' to each side of the equation. -3.428571429z + 2.938775513 + z2 = 0.2857142857 + 2.938775513 Reorder the terms: 2.938775513 + -3.428571429z + z2 = 0.2857142857 + 2.938775513 Combine like terms: 0.2857142857 + 2.938775513 = 3.2244897987 2.938775513 + -3.428571429z + z2 = 3.2244897987 Factor a perfect square on the left side: (z + -1.714285715)(z + -1.714285715) = 3.2244897987 Calculate the square root of the right side: 1.795686442 Break this problem into two subproblems by setting (z + -1.714285715) equal to 1.795686442 and -1.795686442.Subproblem 1
z + -1.714285715 = 1.795686442 Simplifying z + -1.714285715 = 1.795686442 Reorder the terms: -1.714285715 + z = 1.795686442 Solving -1.714285715 + z = 1.795686442 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.714285715' to each side of the equation. -1.714285715 + 1.714285715 + z = 1.795686442 + 1.714285715 Combine like terms: -1.714285715 + 1.714285715 = 0.000000000 0.000000000 + z = 1.795686442 + 1.714285715 z = 1.795686442 + 1.714285715 Combine like terms: 1.795686442 + 1.714285715 = 3.509972157 z = 3.509972157 Simplifying z = 3.509972157Subproblem 2
z + -1.714285715 = -1.795686442 Simplifying z + -1.714285715 = -1.795686442 Reorder the terms: -1.714285715 + z = -1.795686442 Solving -1.714285715 + z = -1.795686442 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.714285715' to each side of the equation. -1.714285715 + 1.714285715 + z = -1.795686442 + 1.714285715 Combine like terms: -1.714285715 + 1.714285715 = 0.000000000 0.000000000 + z = -1.795686442 + 1.714285715 z = -1.795686442 + 1.714285715 Combine like terms: -1.795686442 + 1.714285715 = -0.081400727 z = -0.081400727 Simplifying z = -0.081400727Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.509972157, -0.081400727}
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