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Simplifying 4 + z(2z + 3) = 15 Reorder the terms: 4 + z(3 + 2z) = 15 4 + (3 * z + 2z * z) = 15 4 + (3z + 2z2) = 15 Solving 4 + 3z + 2z2 = 15 Solving for variable 'z'. Reorder the terms: 4 + -15 + 3z + 2z2 = 15 + -15 Combine like terms: 4 + -15 = -11 -11 + 3z + 2z2 = 15 + -15 Combine like terms: 15 + -15 = 0 -11 + 3z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -5.5 + 1.5z + z2 = 0 Move the constant term to the right: Add '5.5' to each side of the equation. -5.5 + 1.5z + 5.5 + z2 = 0 + 5.5 Reorder the terms: -5.5 + 5.5 + 1.5z + z2 = 0 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + 1.5z + z2 = 0 + 5.5 1.5z + z2 = 0 + 5.5 Combine like terms: 0 + 5.5 = 5.5 1.5z + z2 = 5.5 The z term is 1.5z. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5z + 0.5625 + z2 = 5.5 + 0.5625 Reorder the terms: 0.5625 + 1.5z + z2 = 5.5 + 0.5625 Combine like terms: 5.5 + 0.5625 = 6.0625 0.5625 + 1.5z + z2 = 6.0625 Factor a perfect square on the left side: (z + 0.75)(z + 0.75) = 6.0625 Calculate the square root of the right side: 2.46221445 Break this problem into two subproblems by setting (z + 0.75) equal to 2.46221445 and -2.46221445.Subproblem 1
z + 0.75 = 2.46221445 Simplifying z + 0.75 = 2.46221445 Reorder the terms: 0.75 + z = 2.46221445 Solving 0.75 + z = 2.46221445 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + z = 2.46221445 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = 2.46221445 + -0.75 z = 2.46221445 + -0.75 Combine like terms: 2.46221445 + -0.75 = 1.71221445 z = 1.71221445 Simplifying z = 1.71221445Subproblem 2
z + 0.75 = -2.46221445 Simplifying z + 0.75 = -2.46221445 Reorder the terms: 0.75 + z = -2.46221445 Solving 0.75 + z = -2.46221445 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + z = -2.46221445 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = -2.46221445 + -0.75 z = -2.46221445 + -0.75 Combine like terms: -2.46221445 + -0.75 = -3.21221445 z = -3.21221445 Simplifying z = -3.21221445Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.71221445, -3.21221445}
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