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Simplifying 5z2 + -7 = 8z Reorder the terms: -7 + 5z2 = 8z Solving -7 + 5z2 = 8z Solving for variable 'z'. Reorder the terms: -7 + -8z + 5z2 = 8z + -8z Combine like terms: 8z + -8z = 0 -7 + -8z + 5z2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1.4 + -1.6z + z2 = 0 Move the constant term to the right: Add '1.4' to each side of the equation. -1.4 + -1.6z + 1.4 + z2 = 0 + 1.4 Reorder the terms: -1.4 + 1.4 + -1.6z + z2 = 0 + 1.4 Combine like terms: -1.4 + 1.4 = 0.0 0.0 + -1.6z + z2 = 0 + 1.4 -1.6z + z2 = 0 + 1.4 Combine like terms: 0 + 1.4 = 1.4 -1.6z + z2 = 1.4 The z term is -1.6z. Take half its coefficient (-0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. -1.6z + 0.64 + z2 = 1.4 + 0.64 Reorder the terms: 0.64 + -1.6z + z2 = 1.4 + 0.64 Combine like terms: 1.4 + 0.64 = 2.04 0.64 + -1.6z + z2 = 2.04 Factor a perfect square on the left side: (z + -0.8)(z + -0.8) = 2.04 Calculate the square root of the right side: 1.428285686 Break this problem into two subproblems by setting (z + -0.8) equal to 1.428285686 and -1.428285686.Subproblem 1
z + -0.8 = 1.428285686 Simplifying z + -0.8 = 1.428285686 Reorder the terms: -0.8 + z = 1.428285686 Solving -0.8 + z = 1.428285686 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.8' to each side of the equation. -0.8 + 0.8 + z = 1.428285686 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + z = 1.428285686 + 0.8 z = 1.428285686 + 0.8 Combine like terms: 1.428285686 + 0.8 = 2.228285686 z = 2.228285686 Simplifying z = 2.228285686Subproblem 2
z + -0.8 = -1.428285686 Simplifying z + -0.8 = -1.428285686 Reorder the terms: -0.8 + z = -1.428285686 Solving -0.8 + z = -1.428285686 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.8' to each side of the equation. -0.8 + 0.8 + z = -1.428285686 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + z = -1.428285686 + 0.8 z = -1.428285686 + 0.8 Combine like terms: -1.428285686 + 0.8 = -0.628285686 z = -0.628285686 Simplifying z = -0.628285686Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.228285686, -0.628285686}
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