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Simplifying 4z2 + -15z + -6 = 0 Reorder the terms: -6 + -15z + 4z2 = 0 Solving -6 + -15z + 4z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.5 + -3.75z + z2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -3.75z + 1.5 + z2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -3.75z + z2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -3.75z + z2 = 0 + 1.5 -3.75z + z2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -3.75z + z2 = 1.5 The z term is -3.75z. Take half its coefficient (-1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. -3.75z + 3.515625 + z2 = 1.5 + 3.515625 Reorder the terms: 3.515625 + -3.75z + z2 = 1.5 + 3.515625 Combine like terms: 1.5 + 3.515625 = 5.015625 3.515625 + -3.75z + z2 = 5.015625 Factor a perfect square on the left side: (z + -1.875)(z + -1.875) = 5.015625 Calculate the square root of the right side: 2.239559108 Break this problem into two subproblems by setting (z + -1.875) equal to 2.239559108 and -2.239559108.Subproblem 1
z + -1.875 = 2.239559108 Simplifying z + -1.875 = 2.239559108 Reorder the terms: -1.875 + z = 2.239559108 Solving -1.875 + z = 2.239559108 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + z = 2.239559108 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + z = 2.239559108 + 1.875 z = 2.239559108 + 1.875 Combine like terms: 2.239559108 + 1.875 = 4.114559108 z = 4.114559108 Simplifying z = 4.114559108Subproblem 2
z + -1.875 = -2.239559108 Simplifying z + -1.875 = -2.239559108 Reorder the terms: -1.875 + z = -2.239559108 Solving -1.875 + z = -2.239559108 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + z = -2.239559108 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + z = -2.239559108 + 1.875 z = -2.239559108 + 1.875 Combine like terms: -2.239559108 + 1.875 = -0.364559108 z = -0.364559108 Simplifying z = -0.364559108Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.114559108, -0.364559108}
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