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Simplifying 5(4z + 6) + -2(z + -4) = 7z(z + 4) + -2(7z + -2) + -48 Reorder the terms: 5(6 + 4z) + -2(z + -4) = 7z(z + 4) + -2(7z + -2) + -48 (6 * 5 + 4z * 5) + -2(z + -4) = 7z(z + 4) + -2(7z + -2) + -48 (30 + 20z) + -2(z + -4) = 7z(z + 4) + -2(7z + -2) + -48 Reorder the terms: 30 + 20z + -2(-4 + z) = 7z(z + 4) + -2(7z + -2) + -48 30 + 20z + (-4 * -2 + z * -2) = 7z(z + 4) + -2(7z + -2) + -48 30 + 20z + (8 + -2z) = 7z(z + 4) + -2(7z + -2) + -48 Reorder the terms: 30 + 8 + 20z + -2z = 7z(z + 4) + -2(7z + -2) + -48 Combine like terms: 30 + 8 = 38 38 + 20z + -2z = 7z(z + 4) + -2(7z + -2) + -48 Combine like terms: 20z + -2z = 18z 38 + 18z = 7z(z + 4) + -2(7z + -2) + -48 Reorder the terms: 38 + 18z = 7z(4 + z) + -2(7z + -2) + -48 38 + 18z = (4 * 7z + z * 7z) + -2(7z + -2) + -48 38 + 18z = (28z + 7z2) + -2(7z + -2) + -48 Reorder the terms: 38 + 18z = 28z + 7z2 + -2(-2 + 7z) + -48 38 + 18z = 28z + 7z2 + (-2 * -2 + 7z * -2) + -48 38 + 18z = 28z + 7z2 + (4 + -14z) + -48 Reorder the terms: 38 + 18z = 4 + -48 + 28z + -14z + 7z2 Combine like terms: 4 + -48 = -44 38 + 18z = -44 + 28z + -14z + 7z2 Combine like terms: 28z + -14z = 14z 38 + 18z = -44 + 14z + 7z2 Solving 38 + 18z = -44 + 14z + 7z2 Solving for variable 'z'. Reorder the terms: 38 + 44 + 18z + -14z + -7z2 = -44 + 14z + 7z2 + 44 + -14z + -7z2 Combine like terms: 38 + 44 = 82 82 + 18z + -14z + -7z2 = -44 + 14z + 7z2 + 44 + -14z + -7z2 Combine like terms: 18z + -14z = 4z 82 + 4z + -7z2 = -44 + 14z + 7z2 + 44 + -14z + -7z2 Reorder the terms: 82 + 4z + -7z2 = -44 + 44 + 14z + -14z + 7z2 + -7z2 Combine like terms: -44 + 44 = 0 82 + 4z + -7z2 = 0 + 14z + -14z + 7z2 + -7z2 82 + 4z + -7z2 = 14z + -14z + 7z2 + -7z2 Combine like terms: 14z + -14z = 0 82 + 4z + -7z2 = 0 + 7z2 + -7z2 82 + 4z + -7z2 = 7z2 + -7z2 Combine like terms: 7z2 + -7z2 = 0 82 + 4z + -7z2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -11.71428571 + -0.5714285714z + z2 = 0 Move the constant term to the right: Add '11.71428571' to each side of the equation. -11.71428571 + -0.5714285714z + 11.71428571 + z2 = 0 + 11.71428571 Reorder the terms: -11.71428571 + 11.71428571 + -0.5714285714z + z2 = 0 + 11.71428571 Combine like terms: -11.71428571 + 11.71428571 = 0.00000000 0.00000000 + -0.5714285714z + z2 = 0 + 11.71428571 -0.5714285714z + z2 = 0 + 11.71428571 Combine like terms: 0 + 11.71428571 = 11.71428571 -0.5714285714z + z2 = 11.71428571 The z term is -0.5714285714z. Take half its coefficient (-0.2857142857). Square it (0.08163265305) and add it to both sides. Add '0.08163265305' to each side of the equation. -0.5714285714z + 0.08163265305 + z2 = 11.71428571 + 0.08163265305 Reorder the terms: 0.08163265305 + -0.5714285714z + z2 = 11.71428571 + 0.08163265305 Combine like terms: 11.71428571 + 0.08163265305 = 11.79591836305 0.08163265305 + -0.5714285714z + z2 = 11.79591836305 Factor a perfect square on the left side: (z + -0.2857142857)(z + -0.2857142857) = 11.79591836305 Calculate the square root of the right side: 3.434518651 Break this problem into two subproblems by setting (z + -0.2857142857) equal to 3.434518651 and -3.434518651.Subproblem 1
z + -0.2857142857 = 3.434518651 Simplifying z + -0.2857142857 = 3.434518651 Reorder the terms: -0.2857142857 + z = 3.434518651 Solving -0.2857142857 + z = 3.434518651 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + z = 3.434518651 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + z = 3.434518651 + 0.2857142857 z = 3.434518651 + 0.2857142857 Combine like terms: 3.434518651 + 0.2857142857 = 3.7202329367 z = 3.7202329367 Simplifying z = 3.7202329367Subproblem 2
z + -0.2857142857 = -3.434518651 Simplifying z + -0.2857142857 = -3.434518651 Reorder the terms: -0.2857142857 + z = -3.434518651 Solving -0.2857142857 + z = -3.434518651 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + z = -3.434518651 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + z = -3.434518651 + 0.2857142857 z = -3.434518651 + 0.2857142857 Combine like terms: -3.434518651 + 0.2857142857 = -3.1488043653 z = -3.1488043653 Simplifying z = -3.1488043653Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.7202329367, -3.1488043653}
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