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Simplifying 4z2 + -15z + 4 = 0 Reorder the terms: 4 + -15z + 4z2 = 0 Solving 4 + -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 + -3.75z + z2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -3.75z + -1 + z2 = 0 + -1 Reorder the terms: 1 + -1 + -3.75z + z2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -3.75z + z2 = 0 + -1 -3.75z + z2 = 0 + -1 Combine like terms: 0 + -1 = -1 -3.75z + z2 = -1 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 + 3.515625 Reorder the terms: 3.515625 + -3.75z + z2 = -1 + 3.515625 Combine like terms: -1 + 3.515625 = 2.515625 3.515625 + -3.75z + z2 = 2.515625 Factor a perfect square on the left side: (z + -1.875)(z + -1.875) = 2.515625 Calculate the square root of the right side: 1.586072193 Break this problem into two subproblems by setting (z + -1.875) equal to 1.586072193 and -1.586072193.Subproblem 1
z + -1.875 = 1.586072193 Simplifying z + -1.875 = 1.586072193 Reorder the terms: -1.875 + z = 1.586072193 Solving -1.875 + z = 1.586072193 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 = 1.586072193 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + z = 1.586072193 + 1.875 z = 1.586072193 + 1.875 Combine like terms: 1.586072193 + 1.875 = 3.461072193 z = 3.461072193 Simplifying z = 3.461072193Subproblem 2
z + -1.875 = -1.586072193 Simplifying z + -1.875 = -1.586072193 Reorder the terms: -1.875 + z = -1.586072193 Solving -1.875 + z = -1.586072193 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 = -1.586072193 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + z = -1.586072193 + 1.875 z = -1.586072193 + 1.875 Combine like terms: -1.586072193 + 1.875 = 0.288927807 z = 0.288927807 Simplifying z = 0.288927807Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.461072193, 0.288927807}
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