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Simplifying 8z + 3 + -2z2 + 10z = 0 Reorder the terms: 3 + 8z + 10z + -2z2 = 0 Combine like terms: 8z + 10z = 18z 3 + 18z + -2z2 = 0 Solving 3 + 18z + -2z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1.5 + -9z + z2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -9z + 1.5 + z2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -9z + z2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -9z + z2 = 0 + 1.5 -9z + z2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -9z + z2 = 1.5 The z term is -9z. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9z + 20.25 + z2 = 1.5 + 20.25 Reorder the terms: 20.25 + -9z + z2 = 1.5 + 20.25 Combine like terms: 1.5 + 20.25 = 21.75 20.25 + -9z + z2 = 21.75 Factor a perfect square on the left side: (z + -4.5)(z + -4.5) = 21.75 Calculate the square root of the right side: 4.663689527 Break this problem into two subproblems by setting (z + -4.5) equal to 4.663689527 and -4.663689527.Subproblem 1
z + -4.5 = 4.663689527 Simplifying z + -4.5 = 4.663689527 Reorder the terms: -4.5 + z = 4.663689527 Solving -4.5 + z = 4.663689527 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + z = 4.663689527 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + z = 4.663689527 + 4.5 z = 4.663689527 + 4.5 Combine like terms: 4.663689527 + 4.5 = 9.163689527 z = 9.163689527 Simplifying z = 9.163689527Subproblem 2
z + -4.5 = -4.663689527 Simplifying z + -4.5 = -4.663689527 Reorder the terms: -4.5 + z = -4.663689527 Solving -4.5 + z = -4.663689527 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + z = -4.663689527 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + z = -4.663689527 + 4.5 z = -4.663689527 + 4.5 Combine like terms: -4.663689527 + 4.5 = -0.163689527 z = -0.163689527 Simplifying z = -0.163689527Solution
The solution to the problem is based on the solutions from the subproblems. z = {9.163689527, -0.163689527}
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