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Simplifying 5z2 = 7z + 2 Reorder the terms: 5z2 = 2 + 7z Solving 5z2 = 2 + 7z Solving for variable 'z'. Reorder the terms: -2 + -7z + 5z2 = 2 + 7z + -2 + -7z Reorder the terms: -2 + -7z + 5z2 = 2 + -2 + 7z + -7z Combine like terms: 2 + -2 = 0 -2 + -7z + 5z2 = 0 + 7z + -7z -2 + -7z + 5z2 = 7z + -7z Combine like terms: 7z + -7z = 0 -2 + -7z + 5z2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.4 + -1.4z + z2 = 0 Move the constant term to the right: Add '0.4' to each side of the equation. -0.4 + -1.4z + 0.4 + z2 = 0 + 0.4 Reorder the terms: -0.4 + 0.4 + -1.4z + z2 = 0 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + -1.4z + z2 = 0 + 0.4 -1.4z + z2 = 0 + 0.4 Combine like terms: 0 + 0.4 = 0.4 -1.4z + z2 = 0.4 The z term is -1.4z. Take half its coefficient (-0.7). Square it (0.49) and add it to both sides. Add '0.49' to each side of the equation. -1.4z + 0.49 + z2 = 0.4 + 0.49 Reorder the terms: 0.49 + -1.4z + z2 = 0.4 + 0.49 Combine like terms: 0.4 + 0.49 = 0.89 0.49 + -1.4z + z2 = 0.89 Factor a perfect square on the left side: (z + -0.7)(z + -0.7) = 0.89 Calculate the square root of the right side: 0.943398113 Break this problem into two subproblems by setting (z + -0.7) equal to 0.943398113 and -0.943398113.Subproblem 1
z + -0.7 = 0.943398113 Simplifying z + -0.7 = 0.943398113 Reorder the terms: -0.7 + z = 0.943398113 Solving -0.7 + z = 0.943398113 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + z = 0.943398113 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + z = 0.943398113 + 0.7 z = 0.943398113 + 0.7 Combine like terms: 0.943398113 + 0.7 = 1.643398113 z = 1.643398113 Simplifying z = 1.643398113Subproblem 2
z + -0.7 = -0.943398113 Simplifying z + -0.7 = -0.943398113 Reorder the terms: -0.7 + z = -0.943398113 Solving -0.7 + z = -0.943398113 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + z = -0.943398113 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + z = -0.943398113 + 0.7 z = -0.943398113 + 0.7 Combine like terms: -0.943398113 + 0.7 = -0.243398113 z = -0.243398113 Simplifying z = -0.243398113Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.643398113, -0.243398113}
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