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Simplifying 4.25z + 2.5z2 = 5 Solving 4.25z + 2.5z2 = 5 Solving for variable 'z'. Reorder the terms: -5 + 4.25z + 2.5z2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + 4.25z + 2.5z2 = 0 Begin completing the square. Divide all terms by 2.5 the coefficient of the squared term: Divide each side by '2.5'. -2 + 1.7z + z2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 1.7z + 2 + z2 = 0 + 2 Reorder the terms: -2 + 2 + 1.7z + z2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 1.7z + z2 = 0 + 2 1.7z + z2 = 0 + 2 Combine like terms: 0 + 2 = 2 1.7z + z2 = 2 The z term is 1.7z. Take half its coefficient (0.85). Square it (0.7225) and add it to both sides. Add '0.7225' to each side of the equation. 1.7z + 0.7225 + z2 = 2 + 0.7225 Reorder the terms: 0.7225 + 1.7z + z2 = 2 + 0.7225 Combine like terms: 2 + 0.7225 = 2.7225 0.7225 + 1.7z + z2 = 2.7225 Factor a perfect square on the left side: (z + 0.85)(z + 0.85) = 2.7225 Calculate the square root of the right side: 1.65 Break this problem into two subproblems by setting (z + 0.85) equal to 1.65 and -1.65.Subproblem 1
z + 0.85 = 1.65 Simplifying z + 0.85 = 1.65 Reorder the terms: 0.85 + z = 1.65 Solving 0.85 + z = 1.65 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.85' to each side of the equation. 0.85 + -0.85 + z = 1.65 + -0.85 Combine like terms: 0.85 + -0.85 = 0.00 0.00 + z = 1.65 + -0.85 z = 1.65 + -0.85 Combine like terms: 1.65 + -0.85 = 0.8 z = 0.8 Simplifying z = 0.8Subproblem 2
z + 0.85 = -1.65 Simplifying z + 0.85 = -1.65 Reorder the terms: 0.85 + z = -1.65 Solving 0.85 + z = -1.65 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.85' to each side of the equation. 0.85 + -0.85 + z = -1.65 + -0.85 Combine like terms: 0.85 + -0.85 = 0.00 0.00 + z = -1.65 + -0.85 z = -1.65 + -0.85 Combine like terms: -1.65 + -0.85 = -2.5 z = -2.5 Simplifying z = -2.5Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.8, -2.5}
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