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Simplifying 5z2 + -15z = 9 Reorder the terms: -15z + 5z2 = 9 Solving -15z + 5z2 = 9 Solving for variable 'z'. Reorder the terms: -9 + -15z + 5z2 = 9 + -9 Combine like terms: 9 + -9 = 0 -9 + -15z + 5z2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1.8 + -3z + z2 = 0 Move the constant term to the right: Add '1.8' to each side of the equation. -1.8 + -3z + 1.8 + z2 = 0 + 1.8 Reorder the terms: -1.8 + 1.8 + -3z + z2 = 0 + 1.8 Combine like terms: -1.8 + 1.8 = 0.0 0.0 + -3z + z2 = 0 + 1.8 -3z + z2 = 0 + 1.8 Combine like terms: 0 + 1.8 = 1.8 -3z + z2 = 1.8 The z term is -3z. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3z + 2.25 + z2 = 1.8 + 2.25 Reorder the terms: 2.25 + -3z + z2 = 1.8 + 2.25 Combine like terms: 1.8 + 2.25 = 4.05 2.25 + -3z + z2 = 4.05 Factor a perfect square on the left side: (z + -1.5)(z + -1.5) = 4.05 Calculate the square root of the right side: 2.01246118 Break this problem into two subproblems by setting (z + -1.5) equal to 2.01246118 and -2.01246118.Subproblem 1
z + -1.5 = 2.01246118 Simplifying z + -1.5 = 2.01246118 Reorder the terms: -1.5 + z = 2.01246118 Solving -1.5 + z = 2.01246118 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = 2.01246118 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = 2.01246118 + 1.5 z = 2.01246118 + 1.5 Combine like terms: 2.01246118 + 1.5 = 3.51246118 z = 3.51246118 Simplifying z = 3.51246118Subproblem 2
z + -1.5 = -2.01246118 Simplifying z + -1.5 = -2.01246118 Reorder the terms: -1.5 + z = -2.01246118 Solving -1.5 + z = -2.01246118 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = -2.01246118 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = -2.01246118 + 1.5 z = -2.01246118 + 1.5 Combine like terms: -2.01246118 + 1.5 = -0.51246118 z = -0.51246118 Simplifying z = -0.51246118Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.51246118, -0.51246118}
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