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Simplifying 2z2 + -11z = 15 Reorder the terms: -11z + 2z2 = 15 Solving -11z + 2z2 = 15 Solving for variable 'z'. Reorder the terms: -15 + -11z + 2z2 = 15 + -15 Combine like terms: 15 + -15 = 0 -15 + -11z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7.5 + -5.5z + z2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + -5.5z + 7.5 + z2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + -5.5z + z2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + -5.5z + z2 = 0 + 7.5 -5.5z + z2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 -5.5z + z2 = 7.5 The z term is -5.5z. Take half its coefficient (-2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. -5.5z + 7.5625 + z2 = 7.5 + 7.5625 Reorder the terms: 7.5625 + -5.5z + z2 = 7.5 + 7.5625 Combine like terms: 7.5 + 7.5625 = 15.0625 7.5625 + -5.5z + z2 = 15.0625 Factor a perfect square on the left side: (z + -2.75)(z + -2.75) = 15.0625 Calculate the square root of the right side: 3.881043674 Break this problem into two subproblems by setting (z + -2.75) equal to 3.881043674 and -3.881043674.Subproblem 1
z + -2.75 = 3.881043674 Simplifying z + -2.75 = 3.881043674 Reorder the terms: -2.75 + z = 3.881043674 Solving -2.75 + z = 3.881043674 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + z = 3.881043674 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + z = 3.881043674 + 2.75 z = 3.881043674 + 2.75 Combine like terms: 3.881043674 + 2.75 = 6.631043674 z = 6.631043674 Simplifying z = 6.631043674Subproblem 2
z + -2.75 = -3.881043674 Simplifying z + -2.75 = -3.881043674 Reorder the terms: -2.75 + z = -3.881043674 Solving -2.75 + z = -3.881043674 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + z = -3.881043674 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + z = -3.881043674 + 2.75 z = -3.881043674 + 2.75 Combine like terms: -3.881043674 + 2.75 = -1.131043674 z = -1.131043674 Simplifying z = -1.131043674Solution
The solution to the problem is based on the solutions from the subproblems. z = {6.631043674, -1.131043674}
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