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Simplifying 4z2 + 9 = 16z Reorder the terms: 9 + 4z2 = 16z Solving 9 + 4z2 = 16z Solving for variable 'z'. Reorder the terms: 9 + -16z + 4z2 = 16z + -16z Combine like terms: 16z + -16z = 0 9 + -16z + 4z2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 2.25 + -4z + z2 = 0 Move the constant term to the right: Add '-2.25' to each side of the equation. 2.25 + -4z + -2.25 + z2 = 0 + -2.25 Reorder the terms: 2.25 + -2.25 + -4z + z2 = 0 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + -4z + z2 = 0 + -2.25 -4z + z2 = 0 + -2.25 Combine like terms: 0 + -2.25 = -2.25 -4z + z2 = -2.25 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 = -2.25 + 4 Reorder the terms: 4 + -4z + z2 = -2.25 + 4 Combine like terms: -2.25 + 4 = 1.75 4 + -4z + z2 = 1.75 Factor a perfect square on the left side: (z + -2)(z + -2) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (z + -2) equal to 1.322875656 and -1.322875656.Subproblem 1
z + -2 = 1.322875656 Simplifying z + -2 = 1.322875656 Reorder the terms: -2 + z = 1.322875656 Solving -2 + z = 1.322875656 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 = 1.322875656 + 2 Combine like terms: -2 + 2 = 0 0 + z = 1.322875656 + 2 z = 1.322875656 + 2 Combine like terms: 1.322875656 + 2 = 3.322875656 z = 3.322875656 Simplifying z = 3.322875656Subproblem 2
z + -2 = -1.322875656 Simplifying z + -2 = -1.322875656 Reorder the terms: -2 + z = -1.322875656 Solving -2 + z = -1.322875656 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 = -1.322875656 + 2 Combine like terms: -2 + 2 = 0 0 + z = -1.322875656 + 2 z = -1.322875656 + 2 Combine like terms: -1.322875656 + 2 = 0.677124344 z = 0.677124344 Simplifying z = 0.677124344Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.322875656, 0.677124344}
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