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Simplifying 4z2 + 53 = 32z Reorder the terms: 53 + 4z2 = 32z Solving 53 + 4z2 = 32z Solving for variable 'z'. Reorder the terms: 53 + -32z + 4z2 = 32z + -32z Combine like terms: 32z + -32z = 0 53 + -32z + 4z2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 13.25 + -8z + z2 = 0 Move the constant term to the right: Add '-13.25' to each side of the equation. 13.25 + -8z + -13.25 + z2 = 0 + -13.25 Reorder the terms: 13.25 + -13.25 + -8z + z2 = 0 + -13.25 Combine like terms: 13.25 + -13.25 = 0.00 0.00 + -8z + z2 = 0 + -13.25 -8z + z2 = 0 + -13.25 Combine like terms: 0 + -13.25 = -13.25 -8z + z2 = -13.25 The z term is -8z. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8z + 16 + z2 = -13.25 + 16 Reorder the terms: 16 + -8z + z2 = -13.25 + 16 Combine like terms: -13.25 + 16 = 2.75 16 + -8z + z2 = 2.75 Factor a perfect square on the left side: (z + -4)(z + -4) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (z + -4) equal to 1.658312395 and -1.658312395.Subproblem 1
z + -4 = 1.658312395 Simplifying z + -4 = 1.658312395 Reorder the terms: -4 + z = 1.658312395 Solving -4 + z = 1.658312395 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = 1.658312395 + 4 Combine like terms: -4 + 4 = 0 0 + z = 1.658312395 + 4 z = 1.658312395 + 4 Combine like terms: 1.658312395 + 4 = 5.658312395 z = 5.658312395 Simplifying z = 5.658312395Subproblem 2
z + -4 = -1.658312395 Simplifying z + -4 = -1.658312395 Reorder the terms: -4 + z = -1.658312395 Solving -4 + z = -1.658312395 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = -1.658312395 + 4 Combine like terms: -4 + 4 = 0 0 + z = -1.658312395 + 4 z = -1.658312395 + 4 Combine like terms: -1.658312395 + 4 = 2.341687605 z = 2.341687605 Simplifying z = 2.341687605Solution
The solution to the problem is based on the solutions from the subproblems. z = {5.658312395, 2.341687605}
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