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Simplifying -10z + 3(5 + 4z) = z(3z + 3) -10z + (5 * 3 + 4z * 3) = z(3z + 3) -10z + (15 + 12z) = z(3z + 3) Reorder the terms: 15 + -10z + 12z = z(3z + 3) Combine like terms: -10z + 12z = 2z 15 + 2z = z(3z + 3) Reorder the terms: 15 + 2z = z(3 + 3z) 15 + 2z = (3 * z + 3z * z) 15 + 2z = (3z + 3z2) Solving 15 + 2z = 3z + 3z2 Solving for variable 'z'. Combine like terms: 2z + -3z = -1z 15 + -1z + -3z2 = 3z + 3z2 + -3z + -3z2 Reorder the terms: 15 + -1z + -3z2 = 3z + -3z + 3z2 + -3z2 Combine like terms: 3z + -3z = 0 15 + -1z + -3z2 = 0 + 3z2 + -3z2 15 + -1z + -3z2 = 3z2 + -3z2 Combine like terms: 3z2 + -3z2 = 0 15 + -1z + -3z2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -5 + 0.3333333333z + z2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + 0.3333333333z + 5 + z2 = 0 + 5 Reorder the terms: -5 + 5 + 0.3333333333z + z2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 0.3333333333z + z2 = 0 + 5 0.3333333333z + z2 = 0 + 5 Combine like terms: 0 + 5 = 5 0.3333333333z + z2 = 5 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 = 5 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333z + z2 = 5 + 0.02777777779 Combine like terms: 5 + 0.02777777779 = 5.02777777779 0.02777777779 + 0.3333333333z + z2 = 5.02777777779 Factor a perfect square on the left side: (z + 0.1666666667)(z + 0.1666666667) = 5.02777777779 Calculate the square root of the right side: 2.242270675 Break this problem into two subproblems by setting (z + 0.1666666667) equal to 2.242270675 and -2.242270675.Subproblem 1
z + 0.1666666667 = 2.242270675 Simplifying z + 0.1666666667 = 2.242270675 Reorder the terms: 0.1666666667 + z = 2.242270675 Solving 0.1666666667 + z = 2.242270675 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 = 2.242270675 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + z = 2.242270675 + -0.1666666667 z = 2.242270675 + -0.1666666667 Combine like terms: 2.242270675 + -0.1666666667 = 2.0756040083 z = 2.0756040083 Simplifying z = 2.0756040083Subproblem 2
z + 0.1666666667 = -2.242270675 Simplifying z + 0.1666666667 = -2.242270675 Reorder the terms: 0.1666666667 + z = -2.242270675 Solving 0.1666666667 + z = -2.242270675 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 = -2.242270675 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + z = -2.242270675 + -0.1666666667 z = -2.242270675 + -0.1666666667 Combine like terms: -2.242270675 + -0.1666666667 = -2.4089373417 z = -2.4089373417 Simplifying z = -2.4089373417Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.0756040083, -2.4089373417}
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