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Simplifying 4z2 + -20z + 15 = 0 Reorder the terms: 15 + -20z + 4z2 = 0 Solving 15 + -20z + 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'. 3.75 + -5z + z2 = 0 Move the constant term to the right: Add '-3.75' to each side of the equation. 3.75 + -5z + -3.75 + z2 = 0 + -3.75 Reorder the terms: 3.75 + -3.75 + -5z + z2 = 0 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + -5z + z2 = 0 + -3.75 -5z + z2 = 0 + -3.75 Combine like terms: 0 + -3.75 = -3.75 -5z + z2 = -3.75 The z term is -5z. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5z + 6.25 + z2 = -3.75 + 6.25 Reorder the terms: 6.25 + -5z + z2 = -3.75 + 6.25 Combine like terms: -3.75 + 6.25 = 2.5 6.25 + -5z + z2 = 2.5 Factor a perfect square on the left side: (z + -2.5)(z + -2.5) = 2.5 Calculate the square root of the right side: 1.58113883 Break this problem into two subproblems by setting (z + -2.5) equal to 1.58113883 and -1.58113883.Subproblem 1
z + -2.5 = 1.58113883 Simplifying z + -2.5 = 1.58113883 Reorder the terms: -2.5 + z = 1.58113883 Solving -2.5 + z = 1.58113883 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = 1.58113883 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = 1.58113883 + 2.5 z = 1.58113883 + 2.5 Combine like terms: 1.58113883 + 2.5 = 4.08113883 z = 4.08113883 Simplifying z = 4.08113883Subproblem 2
z + -2.5 = -1.58113883 Simplifying z + -2.5 = -1.58113883 Reorder the terms: -2.5 + z = -1.58113883 Solving -2.5 + z = -1.58113883 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = -1.58113883 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = -1.58113883 + 2.5 z = -1.58113883 + 2.5 Combine like terms: -1.58113883 + 2.5 = 0.91886117 z = 0.91886117 Simplifying z = 0.91886117Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.08113883, 0.91886117}
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