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Simplifying -9z2 + 5z + 7 = 0 Reorder the terms: 7 + 5z + -9z2 = 0 Solving 7 + 5z + -9z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -0.7777777778 + -0.5555555556z + z2 = 0 Move the constant term to the right: Add '0.7777777778' to each side of the equation. -0.7777777778 + -0.5555555556z + 0.7777777778 + z2 = 0 + 0.7777777778 Reorder the terms: -0.7777777778 + 0.7777777778 + -0.5555555556z + z2 = 0 + 0.7777777778 Combine like terms: -0.7777777778 + 0.7777777778 = 0.0000000000 0.0000000000 + -0.5555555556z + z2 = 0 + 0.7777777778 -0.5555555556z + z2 = 0 + 0.7777777778 Combine like terms: 0 + 0.7777777778 = 0.7777777778 -0.5555555556z + z2 = 0.7777777778 The z term is -0.5555555556z. Take half its coefficient (-0.2777777778). Square it (0.07716049384) and add it to both sides. Add '0.07716049384' to each side of the equation. -0.5555555556z + 0.07716049384 + z2 = 0.7777777778 + 0.07716049384 Reorder the terms: 0.07716049384 + -0.5555555556z + z2 = 0.7777777778 + 0.07716049384 Combine like terms: 0.7777777778 + 0.07716049384 = 0.85493827164 0.07716049384 + -0.5555555556z + z2 = 0.85493827164 Factor a perfect square on the left side: (z + -0.2777777778)(z + -0.2777777778) = 0.85493827164 Calculate the square root of the right side: 0.924628721 Break this problem into two subproblems by setting (z + -0.2777777778) equal to 0.924628721 and -0.924628721.Subproblem 1
z + -0.2777777778 = 0.924628721 Simplifying z + -0.2777777778 = 0.924628721 Reorder the terms: -0.2777777778 + z = 0.924628721 Solving -0.2777777778 + z = 0.924628721 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.2777777778' to each side of the equation. -0.2777777778 + 0.2777777778 + z = 0.924628721 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + z = 0.924628721 + 0.2777777778 z = 0.924628721 + 0.2777777778 Combine like terms: 0.924628721 + 0.2777777778 = 1.2024064988 z = 1.2024064988 Simplifying z = 1.2024064988Subproblem 2
z + -0.2777777778 = -0.924628721 Simplifying z + -0.2777777778 = -0.924628721 Reorder the terms: -0.2777777778 + z = -0.924628721 Solving -0.2777777778 + z = -0.924628721 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.2777777778' to each side of the equation. -0.2777777778 + 0.2777777778 + z = -0.924628721 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + z = -0.924628721 + 0.2777777778 z = -0.924628721 + 0.2777777778 Combine like terms: -0.924628721 + 0.2777777778 = -0.6468509432 z = -0.6468509432 Simplifying z = -0.6468509432Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.2024064988, -0.6468509432}
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