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Simplifying -9z2 + 4z + 6 = 0 Reorder the terms: 6 + 4z + -9z2 = 0 Solving 6 + 4z + -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'. -0.6666666667 + -0.4444444444z + z2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -0.4444444444z + 0.6666666667 + z2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -0.4444444444z + z2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -0.4444444444z + z2 = 0 + 0.6666666667 -0.4444444444z + z2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -0.4444444444z + z2 = 0.6666666667 The z term is -0.4444444444z. Take half its coefficient (-0.2222222222). Square it (0.04938271604) and add it to both sides. Add '0.04938271604' to each side of the equation. -0.4444444444z + 0.04938271604 + z2 = 0.6666666667 + 0.04938271604 Reorder the terms: 0.04938271604 + -0.4444444444z + z2 = 0.6666666667 + 0.04938271604 Combine like terms: 0.6666666667 + 0.04938271604 = 0.71604938274 0.04938271604 + -0.4444444444z + z2 = 0.71604938274 Factor a perfect square on the left side: (z + -0.2222222222)(z + -0.2222222222) = 0.71604938274 Calculate the square root of the right side: 0.846197012 Break this problem into two subproblems by setting (z + -0.2222222222) equal to 0.846197012 and -0.846197012.Subproblem 1
z + -0.2222222222 = 0.846197012 Simplifying z + -0.2222222222 = 0.846197012 Reorder the terms: -0.2222222222 + z = 0.846197012 Solving -0.2222222222 + z = 0.846197012 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + z = 0.846197012 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + z = 0.846197012 + 0.2222222222 z = 0.846197012 + 0.2222222222 Combine like terms: 0.846197012 + 0.2222222222 = 1.0684192342 z = 1.0684192342 Simplifying z = 1.0684192342Subproblem 2
z + -0.2222222222 = -0.846197012 Simplifying z + -0.2222222222 = -0.846197012 Reorder the terms: -0.2222222222 + z = -0.846197012 Solving -0.2222222222 + z = -0.846197012 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + z = -0.846197012 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + z = -0.846197012 + 0.2222222222 z = -0.846197012 + 0.2222222222 Combine like terms: -0.846197012 + 0.2222222222 = -0.6239747898 z = -0.6239747898 Simplifying z = -0.6239747898Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.0684192342, -0.6239747898}
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