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Simplifying z(2z + 7) = 12 Reorder the terms: z(7 + 2z) = 12 (7 * z + 2z * z) = 12 (7z + 2z2) = 12 Solving 7z + 2z2 = 12 Solving for variable 'z'. Reorder the terms: -12 + 7z + 2z2 = 12 + -12 Combine like terms: 12 + -12 = 0 -12 + 7z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -6 + 3.5z + z2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 3.5z + 6 + z2 = 0 + 6 Reorder the terms: -6 + 6 + 3.5z + z2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 3.5z + z2 = 0 + 6 3.5z + z2 = 0 + 6 Combine like terms: 0 + 6 = 6 3.5z + z2 = 6 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 = 6 + 3.0625 Reorder the terms: 3.0625 + 3.5z + z2 = 6 + 3.0625 Combine like terms: 6 + 3.0625 = 9.0625 3.0625 + 3.5z + z2 = 9.0625 Factor a perfect square on the left side: (z + 1.75)(z + 1.75) = 9.0625 Calculate the square root of the right side: 3.010398645 Break this problem into two subproblems by setting (z + 1.75) equal to 3.010398645 and -3.010398645.Subproblem 1
z + 1.75 = 3.010398645 Simplifying z + 1.75 = 3.010398645 Reorder the terms: 1.75 + z = 3.010398645 Solving 1.75 + z = 3.010398645 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 = 3.010398645 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + z = 3.010398645 + -1.75 z = 3.010398645 + -1.75 Combine like terms: 3.010398645 + -1.75 = 1.260398645 z = 1.260398645 Simplifying z = 1.260398645Subproblem 2
z + 1.75 = -3.010398645 Simplifying z + 1.75 = -3.010398645 Reorder the terms: 1.75 + z = -3.010398645 Solving 1.75 + z = -3.010398645 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 = -3.010398645 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + z = -3.010398645 + -1.75 z = -3.010398645 + -1.75 Combine like terms: -3.010398645 + -1.75 = -4.760398645 z = -4.760398645 Simplifying z = -4.760398645Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.260398645, -4.760398645}
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