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Simplifying 2z2 + 3 = 8z Reorder the terms: 3 + 2z2 = 8z Solving 3 + 2z2 = 8z Solving for variable 'z'. Reorder the terms: 3 + -8z + 2z2 = 8z + -8z Combine like terms: 8z + -8z = 0 3 + -8z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + -4z + z2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -4z + -1.5 + z2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -4z + z2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -4z + z2 = 0 + -1.5 -4z + z2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -4z + z2 = -1.5 The z term is -4z. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4z + 4 + z2 = -1.5 + 4 Reorder the terms: 4 + -4z + z2 = -1.5 + 4 Combine like terms: -1.5 + 4 = 2.5 4 + -4z + z2 = 2.5 Factor a perfect square on the left side: (z + -2)(z + -2) = 2.5 Calculate the square root of the right side: 1.58113883 Break this problem into two subproblems by setting (z + -2) equal to 1.58113883 and -1.58113883.Subproblem 1
z + -2 = 1.58113883 Simplifying z + -2 = 1.58113883 Reorder the terms: -2 + z = 1.58113883 Solving -2 + z = 1.58113883 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = 1.58113883 + 2 Combine like terms: -2 + 2 = 0 0 + z = 1.58113883 + 2 z = 1.58113883 + 2 Combine like terms: 1.58113883 + 2 = 3.58113883 z = 3.58113883 Simplifying z = 3.58113883Subproblem 2
z + -2 = -1.58113883 Simplifying z + -2 = -1.58113883 Reorder the terms: -2 + z = -1.58113883 Solving -2 + z = -1.58113883 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = -1.58113883 + 2 Combine like terms: -2 + 2 = 0 0 + z = -1.58113883 + 2 z = -1.58113883 + 2 Combine like terms: -1.58113883 + 2 = 0.41886117 z = 0.41886117 Simplifying z = 0.41886117Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.58113883, 0.41886117}
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