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Simplifying 11z2 + -10z + -3 = -9z2 + 0.75 Reorder the terms: -3 + -10z + 11z2 = -9z2 + 0.75 Reorder the terms: -3 + -10z + 11z2 = 0.75 + -9z2 Solving -3 + -10z + 11z2 = 0.75 + -9z2 Solving for variable 'z'. Reorder the terms: -3 + -0.75 + -10z + 11z2 + 9z2 = 0.75 + -9z2 + -0.75 + 9z2 Combine like terms: -3 + -0.75 = -3.75 -3.75 + -10z + 11z2 + 9z2 = 0.75 + -9z2 + -0.75 + 9z2 Combine like terms: 11z2 + 9z2 = 20z2 -3.75 + -10z + 20z2 = 0.75 + -9z2 + -0.75 + 9z2 Reorder the terms: -3.75 + -10z + 20z2 = 0.75 + -0.75 + -9z2 + 9z2 Combine like terms: 0.75 + -0.75 = 0.00 -3.75 + -10z + 20z2 = 0.00 + -9z2 + 9z2 -3.75 + -10z + 20z2 = -9z2 + 9z2 Combine like terms: -9z2 + 9z2 = 0 -3.75 + -10z + 20z2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -0.1875 + -0.5z + z2 = 0.0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -0.5z + 0.1875 + z2 = 0.0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -0.5z + z2 = 0.0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -0.5z + z2 = 0.0 + 0.1875 -0.5z + z2 = 0.0 + 0.1875 Combine like terms: 0.0 + 0.1875 = 0.1875 -0.5z + z2 = 0.1875 The z term is -0.5z. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5z + 0.0625 + z2 = 0.1875 + 0.0625 Reorder the terms: 0.0625 + -0.5z + z2 = 0.1875 + 0.0625 Combine like terms: 0.1875 + 0.0625 = 0.25 0.0625 + -0.5z + z2 = 0.25 Factor a perfect square on the left side: (z + -0.25)(z + -0.25) = 0.25 Calculate the square root of the right side: 0.5 Break this problem into two subproblems by setting (z + -0.25) equal to 0.5 and -0.5.Subproblem 1
z + -0.25 = 0.5 Simplifying z + -0.25 = 0.5 Reorder the terms: -0.25 + z = 0.5 Solving -0.25 + z = 0.5 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + z = 0.5 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + z = 0.5 + 0.25 z = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 z = 0.75 Simplifying z = 0.75Subproblem 2
z + -0.25 = -0.5 Simplifying z + -0.25 = -0.5 Reorder the terms: -0.25 + z = -0.5 Solving -0.25 + z = -0.5 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + z = -0.5 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + z = -0.5 + 0.25 z = -0.5 + 0.25 Combine like terms: -0.5 + 0.25 = -0.25 z = -0.25 Simplifying z = -0.25Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.75, -0.25}
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