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Simplifying z(2z + 7) = 7 Reorder the terms: z(7 + 2z) = 7 (7 * z + 2z * z) = 7 (7z + 2z2) = 7 Solving 7z + 2z2 = 7 Solving for variable 'z'. Reorder the terms: -7 + 7z + 2z2 = 7 + -7 Combine like terms: 7 + -7 = 0 -7 + 7z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + 3.5z + z2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 3.5z + 3.5 + z2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 3.5z + z2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 3.5z + z2 = 0 + 3.5 3.5z + z2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 3.5z + z2 = 3.5 The z term is 3.5z. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5z + 3.0625 + z2 = 3.5 + 3.0625 Reorder the terms: 3.0625 + 3.5z + z2 = 3.5 + 3.0625 Combine like terms: 3.5 + 3.0625 = 6.5625 3.0625 + 3.5z + z2 = 6.5625 Factor a perfect square on the left side: (z + 1.75)(z + 1.75) = 6.5625 Calculate the square root of the right side: 2.561737691 Break this problem into two subproblems by setting (z + 1.75) equal to 2.561737691 and -2.561737691.Subproblem 1
z + 1.75 = 2.561737691 Simplifying z + 1.75 = 2.561737691 Reorder the terms: 1.75 + z = 2.561737691 Solving 1.75 + z = 2.561737691 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + z = 2.561737691 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + z = 2.561737691 + -1.75 z = 2.561737691 + -1.75 Combine like terms: 2.561737691 + -1.75 = 0.811737691 z = 0.811737691 Simplifying z = 0.811737691Subproblem 2
z + 1.75 = -2.561737691 Simplifying z + 1.75 = -2.561737691 Reorder the terms: 1.75 + z = -2.561737691 Solving 1.75 + z = -2.561737691 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + z = -2.561737691 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + z = -2.561737691 + -1.75 z = -2.561737691 + -1.75 Combine like terms: -2.561737691 + -1.75 = -4.311737691 z = -4.311737691 Simplifying z = -4.311737691Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.811737691, -4.311737691}
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