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Simplifying 9z2 + -54z + 79 = 0 Reorder the terms: 79 + -54z + 9z2 = 0 Solving 79 + -54z + 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'. 8.777777778 + -6z + z2 = 0 Move the constant term to the right: Add '-8.777777778' to each side of the equation. 8.777777778 + -6z + -8.777777778 + z2 = 0 + -8.777777778 Reorder the terms: 8.777777778 + -8.777777778 + -6z + z2 = 0 + -8.777777778 Combine like terms: 8.777777778 + -8.777777778 = 0.000000000 0.000000000 + -6z + z2 = 0 + -8.777777778 -6z + z2 = 0 + -8.777777778 Combine like terms: 0 + -8.777777778 = -8.777777778 -6z + z2 = -8.777777778 The z term is -6z. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6z + 9 + z2 = -8.777777778 + 9 Reorder the terms: 9 + -6z + z2 = -8.777777778 + 9 Combine like terms: -8.777777778 + 9 = 0.222222222 9 + -6z + z2 = 0.222222222 Factor a perfect square on the left side: (z + -3)(z + -3) = 0.222222222 Calculate the square root of the right side: 0.471404521 Break this problem into two subproblems by setting (z + -3) equal to 0.471404521 and -0.471404521.Subproblem 1
z + -3 = 0.471404521 Simplifying z + -3 = 0.471404521 Reorder the terms: -3 + z = 0.471404521 Solving -3 + z = 0.471404521 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = 0.471404521 + 3 Combine like terms: -3 + 3 = 0 0 + z = 0.471404521 + 3 z = 0.471404521 + 3 Combine like terms: 0.471404521 + 3 = 3.471404521 z = 3.471404521 Simplifying z = 3.471404521Subproblem 2
z + -3 = -0.471404521 Simplifying z + -3 = -0.471404521 Reorder the terms: -3 + z = -0.471404521 Solving -3 + z = -0.471404521 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = -0.471404521 + 3 Combine like terms: -3 + 3 = 0 0 + z = -0.471404521 + 3 z = -0.471404521 + 3 Combine like terms: -0.471404521 + 3 = 2.528595479 z = 2.528595479 Simplifying z = 2.528595479Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.471404521, 2.528595479}
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