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Simplifying 3z(z + 4) = (z2 + 9) Reorder the terms: 3z(4 + z) = (z2 + 9) (4 * 3z + z * 3z) = (z2 + 9) (12z + 3z2) = (z2 + 9) Reorder the terms: 12z + 3z2 = (9 + z2) Remove parenthesis around (9 + z2) 12z + 3z2 = 9 + z2 Solving 12z + 3z2 = 9 + z2 Solving for variable 'z'. Reorder the terms: -9 + 12z + 3z2 + -1z2 = 9 + z2 + -9 + -1z2 Combine like terms: 3z2 + -1z2 = 2z2 -9 + 12z + 2z2 = 9 + z2 + -9 + -1z2 Reorder the terms: -9 + 12z + 2z2 = 9 + -9 + z2 + -1z2 Combine like terms: 9 + -9 = 0 -9 + 12z + 2z2 = 0 + z2 + -1z2 -9 + 12z + 2z2 = z2 + -1z2 Combine like terms: z2 + -1z2 = 0 -9 + 12z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4.5 + 6z + z2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + 6z + 4.5 + z2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + 6z + z2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + 6z + z2 = 0 + 4.5 6z + z2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 6z + z2 = 4.5 The z term is 6z. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6z + 9 + z2 = 4.5 + 9 Reorder the terms: 9 + 6z + z2 = 4.5 + 9 Combine like terms: 4.5 + 9 = 13.5 9 + 6z + z2 = 13.5 Factor a perfect square on the left side: (z + 3)(z + 3) = 13.5 Calculate the square root of the right side: 3.674234614 Break this problem into two subproblems by setting (z + 3) equal to 3.674234614 and -3.674234614.Subproblem 1
z + 3 = 3.674234614 Simplifying z + 3 = 3.674234614 Reorder the terms: 3 + z = 3.674234614 Solving 3 + z = 3.674234614 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + z = 3.674234614 + -3 Combine like terms: 3 + -3 = 0 0 + z = 3.674234614 + -3 z = 3.674234614 + -3 Combine like terms: 3.674234614 + -3 = 0.674234614 z = 0.674234614 Simplifying z = 0.674234614Subproblem 2
z + 3 = -3.674234614 Simplifying z + 3 = -3.674234614 Reorder the terms: 3 + z = -3.674234614 Solving 3 + z = -3.674234614 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + z = -3.674234614 + -3 Combine like terms: 3 + -3 = 0 0 + z = -3.674234614 + -3 z = -3.674234614 + -3 Combine like terms: -3.674234614 + -3 = -6.674234614 z = -6.674234614 Simplifying z = -6.674234614Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.674234614, -6.674234614}
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