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Simplifying 7z2 + -18z = 19 Reorder the terms: -18z + 7z2 = 19 Solving -18z + 7z2 = 19 Solving for variable 'z'. Reorder the terms: -19 + -18z + 7z2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + -18z + 7z2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2.714285714 + -2.571428571z + z2 = 0 Move the constant term to the right: Add '2.714285714' to each side of the equation. -2.714285714 + -2.571428571z + 2.714285714 + z2 = 0 + 2.714285714 Reorder the terms: -2.714285714 + 2.714285714 + -2.571428571z + z2 = 0 + 2.714285714 Combine like terms: -2.714285714 + 2.714285714 = 0.000000000 0.000000000 + -2.571428571z + z2 = 0 + 2.714285714 -2.571428571z + z2 = 0 + 2.714285714 Combine like terms: 0 + 2.714285714 = 2.714285714 -2.571428571z + z2 = 2.714285714 The z term is -2.571428571z. Take half its coefficient (-1.285714286). Square it (1.653061225) and add it to both sides. Add '1.653061225' to each side of the equation. -2.571428571z + 1.653061225 + z2 = 2.714285714 + 1.653061225 Reorder the terms: 1.653061225 + -2.571428571z + z2 = 2.714285714 + 1.653061225 Combine like terms: 2.714285714 + 1.653061225 = 4.367346939 1.653061225 + -2.571428571z + z2 = 4.367346939 Factor a perfect square on the left side: (z + -1.285714286)(z + -1.285714286) = 4.367346939 Calculate the square root of the right side: 2.089819834 Break this problem into two subproblems by setting (z + -1.285714286) equal to 2.089819834 and -2.089819834.Subproblem 1
z + -1.285714286 = 2.089819834 Simplifying z + -1.285714286 = 2.089819834 Reorder the terms: -1.285714286 + z = 2.089819834 Solving -1.285714286 + z = 2.089819834 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.285714286' to each side of the equation. -1.285714286 + 1.285714286 + z = 2.089819834 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + z = 2.089819834 + 1.285714286 z = 2.089819834 + 1.285714286 Combine like terms: 2.089819834 + 1.285714286 = 3.37553412 z = 3.37553412 Simplifying z = 3.37553412Subproblem 2
z + -1.285714286 = -2.089819834 Simplifying z + -1.285714286 = -2.089819834 Reorder the terms: -1.285714286 + z = -2.089819834 Solving -1.285714286 + z = -2.089819834 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.285714286' to each side of the equation. -1.285714286 + 1.285714286 + z = -2.089819834 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + z = -2.089819834 + 1.285714286 z = -2.089819834 + 1.285714286 Combine like terms: -2.089819834 + 1.285714286 = -0.804105548 z = -0.804105548 Simplifying z = -0.804105548Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.37553412, -0.804105548}
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