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Simplifying 7z2 + -11z + -13 = 12 + -9z Reorder the terms: -13 + -11z + 7z2 = 12 + -9z Solving -13 + -11z + 7z2 = 12 + -9z Solving for variable 'z'. Reorder the terms: -13 + -12 + -11z + 9z + 7z2 = 12 + -9z + -12 + 9z Combine like terms: -13 + -12 = -25 -25 + -11z + 9z + 7z2 = 12 + -9z + -12 + 9z Combine like terms: -11z + 9z = -2z -25 + -2z + 7z2 = 12 + -9z + -12 + 9z Reorder the terms: -25 + -2z + 7z2 = 12 + -12 + -9z + 9z Combine like terms: 12 + -12 = 0 -25 + -2z + 7z2 = 0 + -9z + 9z -25 + -2z + 7z2 = -9z + 9z Combine like terms: -9z + 9z = 0 -25 + -2z + 7z2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -3.571428571 + -0.2857142857z + z2 = 0 Move the constant term to the right: Add '3.571428571' to each side of the equation. -3.571428571 + -0.2857142857z + 3.571428571 + z2 = 0 + 3.571428571 Reorder the terms: -3.571428571 + 3.571428571 + -0.2857142857z + z2 = 0 + 3.571428571 Combine like terms: -3.571428571 + 3.571428571 = 0.000000000 0.000000000 + -0.2857142857z + z2 = 0 + 3.571428571 -0.2857142857z + z2 = 0 + 3.571428571 Combine like terms: 0 + 3.571428571 = 3.571428571 -0.2857142857z + z2 = 3.571428571 The z term is -0.2857142857z. Take half its coefficient (-0.1428571429). Square it (0.02040816328) and add it to both sides. Add '0.02040816328' to each side of the equation. -0.2857142857z + 0.02040816328 + z2 = 3.571428571 + 0.02040816328 Reorder the terms: 0.02040816328 + -0.2857142857z + z2 = 3.571428571 + 0.02040816328 Combine like terms: 3.571428571 + 0.02040816328 = 3.59183673428 0.02040816328 + -0.2857142857z + z2 = 3.59183673428 Factor a perfect square on the left side: (z + -0.1428571429)(z + -0.1428571429) = 3.59183673428 Calculate the square root of the right side: 1.895214166 Break this problem into two subproblems by setting (z + -0.1428571429) equal to 1.895214166 and -1.895214166.Subproblem 1
z + -0.1428571429 = 1.895214166 Simplifying z + -0.1428571429 = 1.895214166 Reorder the terms: -0.1428571429 + z = 1.895214166 Solving -0.1428571429 + z = 1.895214166 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + z = 1.895214166 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + z = 1.895214166 + 0.1428571429 z = 1.895214166 + 0.1428571429 Combine like terms: 1.895214166 + 0.1428571429 = 2.0380713089 z = 2.0380713089 Simplifying z = 2.0380713089Subproblem 2
z + -0.1428571429 = -1.895214166 Simplifying z + -0.1428571429 = -1.895214166 Reorder the terms: -0.1428571429 + z = -1.895214166 Solving -0.1428571429 + z = -1.895214166 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + z = -1.895214166 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + z = -1.895214166 + 0.1428571429 z = -1.895214166 + 0.1428571429 Combine like terms: -1.895214166 + 0.1428571429 = -1.7523570231 z = -1.7523570231 Simplifying z = -1.7523570231Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.0380713089, -1.7523570231}
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