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Simplifying 2z2 + 7z + 5 = 6z2 Reorder the terms: 5 + 7z + 2z2 = 6z2 Solving 5 + 7z + 2z2 = 6z2 Solving for variable 'z'. Combine like terms: 2z2 + -6z2 = -4z2 5 + 7z + -4z2 = 6z2 + -6z2 Combine like terms: 6z2 + -6z2 = 0 5 + 7z + -4z2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1.25 + -1.75z + z2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -1.75z + 1.25 + z2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -1.75z + z2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -1.75z + z2 = 0 + 1.25 -1.75z + z2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -1.75z + z2 = 1.25 The z term is -1.75z. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75z + 0.765625 + z2 = 1.25 + 0.765625 Reorder the terms: 0.765625 + -1.75z + z2 = 1.25 + 0.765625 Combine like terms: 1.25 + 0.765625 = 2.015625 0.765625 + -1.75z + z2 = 2.015625 Factor a perfect square on the left side: (z + -0.875)(z + -0.875) = 2.015625 Calculate the square root of the right side: 1.419727086 Break this problem into two subproblems by setting (z + -0.875) equal to 1.419727086 and -1.419727086.Subproblem 1
z + -0.875 = 1.419727086 Simplifying z + -0.875 = 1.419727086 Reorder the terms: -0.875 + z = 1.419727086 Solving -0.875 + z = 1.419727086 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + z = 1.419727086 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + z = 1.419727086 + 0.875 z = 1.419727086 + 0.875 Combine like terms: 1.419727086 + 0.875 = 2.294727086 z = 2.294727086 Simplifying z = 2.294727086Subproblem 2
z + -0.875 = -1.419727086 Simplifying z + -0.875 = -1.419727086 Reorder the terms: -0.875 + z = -1.419727086 Solving -0.875 + z = -1.419727086 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + z = -1.419727086 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + z = -1.419727086 + 0.875 z = -1.419727086 + 0.875 Combine like terms: -1.419727086 + 0.875 = -0.544727086 z = -0.544727086 Simplifying z = -0.544727086Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.294727086, -0.544727086}
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