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Simplifying (z + 1)(3z + -2) = 2 + 5 Reorder the terms: (1 + z)(3z + -2) = 2 + 5 Reorder the terms: (1 + z)(-2 + 3z) = 2 + 5 Multiply (1 + z) * (-2 + 3z) (1(-2 + 3z) + z(-2 + 3z)) = 2 + 5 ((-2 * 1 + 3z * 1) + z(-2 + 3z)) = 2 + 5 ((-2 + 3z) + z(-2 + 3z)) = 2 + 5 (-2 + 3z + (-2 * z + 3z * z)) = 2 + 5 (-2 + 3z + (-2z + 3z2)) = 2 + 5 Combine like terms: 3z + -2z = 1z (-2 + 1z + 3z2) = 2 + 5 Combine like terms: 2 + 5 = 7 -2 + 1z + 3z2 = 7 Solving -2 + 1z + 3z2 = 7 Solving for variable 'z'. Reorder the terms: -2 + -7 + 1z + 3z2 = 7 + -7 Combine like terms: -2 + -7 = -9 -9 + 1z + 3z2 = 7 + -7 Combine like terms: 7 + -7 = 0 -9 + 1z + 3z2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -3 + 0.3333333333z + z2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 0.3333333333z + 3 + z2 = 0 + 3 Reorder the terms: -3 + 3 + 0.3333333333z + z2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 0.3333333333z + z2 = 0 + 3 0.3333333333z + z2 = 0 + 3 Combine like terms: 0 + 3 = 3 0.3333333333z + z2 = 3 The z term is 0.3333333333z. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333z + 0.02777777779 + z2 = 3 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333z + z2 = 3 + 0.02777777779 Combine like terms: 3 + 0.02777777779 = 3.02777777779 0.02777777779 + 0.3333333333z + z2 = 3.02777777779 Factor a perfect square on the left side: (z + 0.1666666667)(z + 0.1666666667) = 3.02777777779 Calculate the square root of the right side: 1.740051085 Break this problem into two subproblems by setting (z + 0.1666666667) equal to 1.740051085 and -1.740051085.Subproblem 1
z + 0.1666666667 = 1.740051085 Simplifying z + 0.1666666667 = 1.740051085 Reorder the terms: 0.1666666667 + z = 1.740051085 Solving 0.1666666667 + z = 1.740051085 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + z = 1.740051085 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + z = 1.740051085 + -0.1666666667 z = 1.740051085 + -0.1666666667 Combine like terms: 1.740051085 + -0.1666666667 = 1.5733844183 z = 1.5733844183 Simplifying z = 1.5733844183Subproblem 2
z + 0.1666666667 = -1.740051085 Simplifying z + 0.1666666667 = -1.740051085 Reorder the terms: 0.1666666667 + z = -1.740051085 Solving 0.1666666667 + z = -1.740051085 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + z = -1.740051085 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + z = -1.740051085 + -0.1666666667 z = -1.740051085 + -0.1666666667 Combine like terms: -1.740051085 + -0.1666666667 = -1.9067177517 z = -1.9067177517 Simplifying z = -1.9067177517Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.5733844183, -1.9067177517}
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