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Simplifying 8z * 7z = -9z + 7 Reorder the terms for easier multiplication: 8 * 7z * z = -9z + 7 Multiply 8 * 7 56z * z = -9z + 7 Multiply z * z 56z2 = -9z + 7 Reorder the terms: 56z2 = 7 + -9z Solving 56z2 = 7 + -9z Solving for variable 'z'. Reorder the terms: -7 + 9z + 56z2 = 7 + -9z + -7 + 9z Reorder the terms: -7 + 9z + 56z2 = 7 + -7 + -9z + 9z Combine like terms: 7 + -7 = 0 -7 + 9z + 56z2 = 0 + -9z + 9z -7 + 9z + 56z2 = -9z + 9z Combine like terms: -9z + 9z = 0 -7 + 9z + 56z2 = 0 Begin completing the square. Divide all terms by 56 the coefficient of the squared term: Divide each side by '56'. -0.125 + 0.1607142857z + z2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + 0.1607142857z + 0.125 + z2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + 0.1607142857z + z2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + 0.1607142857z + z2 = 0 + 0.125 0.1607142857z + z2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 0.1607142857z + z2 = 0.125 The z term is 0.1607142857z. Take half its coefficient (0.08035714285). Square it (0.006457270407) and add it to both sides. Add '0.006457270407' to each side of the equation. 0.1607142857z + 0.006457270407 + z2 = 0.125 + 0.006457270407 Reorder the terms: 0.006457270407 + 0.1607142857z + z2 = 0.125 + 0.006457270407 Combine like terms: 0.125 + 0.006457270407 = 0.131457270407 0.006457270407 + 0.1607142857z + z2 = 0.131457270407 Factor a perfect square on the left side: (z + 0.08035714285)(z + 0.08035714285) = 0.131457270407 Calculate the square root of the right side: 0.362570366 Break this problem into two subproblems by setting (z + 0.08035714285) equal to 0.362570366 and -0.362570366.Subproblem 1
z + 0.08035714285 = 0.362570366 Simplifying z + 0.08035714285 = 0.362570366 Reorder the terms: 0.08035714285 + z = 0.362570366 Solving 0.08035714285 + z = 0.362570366 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.08035714285' to each side of the equation. 0.08035714285 + -0.08035714285 + z = 0.362570366 + -0.08035714285 Combine like terms: 0.08035714285 + -0.08035714285 = 0.00000000000 0.00000000000 + z = 0.362570366 + -0.08035714285 z = 0.362570366 + -0.08035714285 Combine like terms: 0.362570366 + -0.08035714285 = 0.28221322315 z = 0.28221322315 Simplifying z = 0.28221322315Subproblem 2
z + 0.08035714285 = -0.362570366 Simplifying z + 0.08035714285 = -0.362570366 Reorder the terms: 0.08035714285 + z = -0.362570366 Solving 0.08035714285 + z = -0.362570366 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.08035714285' to each side of the equation. 0.08035714285 + -0.08035714285 + z = -0.362570366 + -0.08035714285 Combine like terms: 0.08035714285 + -0.08035714285 = 0.00000000000 0.00000000000 + z = -0.362570366 + -0.08035714285 z = -0.362570366 + -0.08035714285 Combine like terms: -0.362570366 + -0.08035714285 = -0.44292750885 z = -0.44292750885 Simplifying z = -0.44292750885Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.28221322315, -0.44292750885}
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