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Simplifying 16z2 + 61 = 64z Reorder the terms: 61 + 16z2 = 64z Solving 61 + 16z2 = 64z Solving for variable 'z'. Reorder the terms: 61 + -64z + 16z2 = 64z + -64z Combine like terms: 64z + -64z = 0 61 + -64z + 16z2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 3.8125 + -4z + z2 = 0 Move the constant term to the right: Add '-3.8125' to each side of the equation. 3.8125 + -4z + -3.8125 + z2 = 0 + -3.8125 Reorder the terms: 3.8125 + -3.8125 + -4z + z2 = 0 + -3.8125 Combine like terms: 3.8125 + -3.8125 = 0.0000 0.0000 + -4z + z2 = 0 + -3.8125 -4z + z2 = 0 + -3.8125 Combine like terms: 0 + -3.8125 = -3.8125 -4z + z2 = -3.8125 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 = -3.8125 + 4 Reorder the terms: 4 + -4z + z2 = -3.8125 + 4 Combine like terms: -3.8125 + 4 = 0.1875 4 + -4z + z2 = 0.1875 Factor a perfect square on the left side: (z + -2)(z + -2) = 0.1875 Calculate the square root of the right side: 0.433012702 Break this problem into two subproblems by setting (z + -2) equal to 0.433012702 and -0.433012702.Subproblem 1
z + -2 = 0.433012702 Simplifying z + -2 = 0.433012702 Reorder the terms: -2 + z = 0.433012702 Solving -2 + z = 0.433012702 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 = 0.433012702 + 2 Combine like terms: -2 + 2 = 0 0 + z = 0.433012702 + 2 z = 0.433012702 + 2 Combine like terms: 0.433012702 + 2 = 2.433012702 z = 2.433012702 Simplifying z = 2.433012702Subproblem 2
z + -2 = -0.433012702 Simplifying z + -2 = -0.433012702 Reorder the terms: -2 + z = -0.433012702 Solving -2 + z = -0.433012702 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 = -0.433012702 + 2 Combine like terms: -2 + 2 = 0 0 + z = -0.433012702 + 2 z = -0.433012702 + 2 Combine like terms: -0.433012702 + 2 = 1.566987298 z = 1.566987298 Simplifying z = 1.566987298Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.433012702, 1.566987298}
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