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Simplifying 9z(z + -5) = 1 Reorder the terms: 9z(-5 + z) = 1 (-5 * 9z + z * 9z) = 1 (-45z + 9z2) = 1 Solving -45z + 9z2 = 1 Solving for variable 'z'. Reorder the terms: -1 + -45z + 9z2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -45z + 9z2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.1111111111 + -5z + z2 = 0 Move the constant term to the right: Add '0.1111111111' to each side of the equation. -0.1111111111 + -5z + 0.1111111111 + z2 = 0 + 0.1111111111 Reorder the terms: -0.1111111111 + 0.1111111111 + -5z + z2 = 0 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + -5z + z2 = 0 + 0.1111111111 -5z + z2 = 0 + 0.1111111111 Combine like terms: 0 + 0.1111111111 = 0.1111111111 -5z + z2 = 0.1111111111 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 = 0.1111111111 + 6.25 Reorder the terms: 6.25 + -5z + z2 = 0.1111111111 + 6.25 Combine like terms: 0.1111111111 + 6.25 = 6.3611111111 6.25 + -5z + z2 = 6.3611111111 Factor a perfect square on the left side: (z + -2.5)(z + -2.5) = 6.3611111111 Calculate the square root of the right side: 2.522124325 Break this problem into two subproblems by setting (z + -2.5) equal to 2.522124325 and -2.522124325.Subproblem 1
z + -2.5 = 2.522124325 Simplifying z + -2.5 = 2.522124325 Reorder the terms: -2.5 + z = 2.522124325 Solving -2.5 + z = 2.522124325 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 = 2.522124325 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = 2.522124325 + 2.5 z = 2.522124325 + 2.5 Combine like terms: 2.522124325 + 2.5 = 5.022124325 z = 5.022124325 Simplifying z = 5.022124325Subproblem 2
z + -2.5 = -2.522124325 Simplifying z + -2.5 = -2.522124325 Reorder the terms: -2.5 + z = -2.522124325 Solving -2.5 + z = -2.522124325 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 = -2.522124325 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = -2.522124325 + 2.5 z = -2.522124325 + 2.5 Combine like terms: -2.522124325 + 2.5 = -0.022124325 z = -0.022124325 Simplifying z = -0.022124325Solution
The solution to the problem is based on the solutions from the subproblems. z = {5.022124325, -0.022124325}
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