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Simplifying (3z + -2)(8 + z) = 2 Reorder the terms: (-2 + 3z)(8 + z) = 2 Multiply (-2 + 3z) * (8 + z) (-2(8 + z) + 3z * (8 + z)) = 2 ((8 * -2 + z * -2) + 3z * (8 + z)) = 2 ((-16 + -2z) + 3z * (8 + z)) = 2 (-16 + -2z + (8 * 3z + z * 3z)) = 2 (-16 + -2z + (24z + 3z2)) = 2 Combine like terms: -2z + 24z = 22z (-16 + 22z + 3z2) = 2 Solving -16 + 22z + 3z2 = 2 Solving for variable 'z'. Reorder the terms: -16 + -2 + 22z + 3z2 = 2 + -2 Combine like terms: -16 + -2 = -18 -18 + 22z + 3z2 = 2 + -2 Combine like terms: 2 + -2 = 0 -18 + 22z + 3z2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -6 + 7.333333333z + z2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 7.333333333z + 6 + z2 = 0 + 6 Reorder the terms: -6 + 6 + 7.333333333z + z2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 7.333333333z + z2 = 0 + 6 7.333333333z + z2 = 0 + 6 Combine like terms: 0 + 6 = 6 7.333333333z + z2 = 6 The z term is 7.333333333z. Take half its coefficient (3.666666667). Square it (13.44444445) and add it to both sides. Add '13.44444445' to each side of the equation. 7.333333333z + 13.44444445 + z2 = 6 + 13.44444445 Reorder the terms: 13.44444445 + 7.333333333z + z2 = 6 + 13.44444445 Combine like terms: 6 + 13.44444445 = 19.44444445 13.44444445 + 7.333333333z + z2 = 19.44444445 Factor a perfect square on the left side: (z + 3.666666667)(z + 3.666666667) = 19.44444445 Calculate the square root of the right side: 4.409585519 Break this problem into two subproblems by setting (z + 3.666666667) equal to 4.409585519 and -4.409585519.Subproblem 1
z + 3.666666667 = 4.409585519 Simplifying z + 3.666666667 = 4.409585519 Reorder the terms: 3.666666667 + z = 4.409585519 Solving 3.666666667 + z = 4.409585519 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.666666667' to each side of the equation. 3.666666667 + -3.666666667 + z = 4.409585519 + -3.666666667 Combine like terms: 3.666666667 + -3.666666667 = 0.000000000 0.000000000 + z = 4.409585519 + -3.666666667 z = 4.409585519 + -3.666666667 Combine like terms: 4.409585519 + -3.666666667 = 0.742918852 z = 0.742918852 Simplifying z = 0.742918852Subproblem 2
z + 3.666666667 = -4.409585519 Simplifying z + 3.666666667 = -4.409585519 Reorder the terms: 3.666666667 + z = -4.409585519 Solving 3.666666667 + z = -4.409585519 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.666666667' to each side of the equation. 3.666666667 + -3.666666667 + z = -4.409585519 + -3.666666667 Combine like terms: 3.666666667 + -3.666666667 = 0.000000000 0.000000000 + z = -4.409585519 + -3.666666667 z = -4.409585519 + -3.666666667 Combine like terms: -4.409585519 + -3.666666667 = -8.076252186 z = -8.076252186 Simplifying z = -8.076252186Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.742918852, -8.076252186}
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