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Simplifying 7z2 + -12z = 20 Reorder the terms: -12z + 7z2 = 20 Solving -12z + 7z2 = 20 Solving for variable 'z'. Reorder the terms: -20 + -12z + 7z2 = 20 + -20 Combine like terms: 20 + -20 = 0 -20 + -12z + 7z2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2.857142857 + -1.714285714z + z2 = 0 Move the constant term to the right: Add '2.857142857' to each side of the equation. -2.857142857 + -1.714285714z + 2.857142857 + z2 = 0 + 2.857142857 Reorder the terms: -2.857142857 + 2.857142857 + -1.714285714z + z2 = 0 + 2.857142857 Combine like terms: -2.857142857 + 2.857142857 = 0.000000000 0.000000000 + -1.714285714z + z2 = 0 + 2.857142857 -1.714285714z + z2 = 0 + 2.857142857 Combine like terms: 0 + 2.857142857 = 2.857142857 -1.714285714z + z2 = 2.857142857 The z term is -1.714285714z. Take half its coefficient (-0.857142857). Square it (0.7346938773) and add it to both sides. Add '0.7346938773' to each side of the equation. -1.714285714z + 0.7346938773 + z2 = 2.857142857 + 0.7346938773 Reorder the terms: 0.7346938773 + -1.714285714z + z2 = 2.857142857 + 0.7346938773 Combine like terms: 2.857142857 + 0.7346938773 = 3.5918367343 0.7346938773 + -1.714285714z + z2 = 3.5918367343 Factor a perfect square on the left side: (z + -0.857142857)(z + -0.857142857) = 3.5918367343 Calculate the square root of the right side: 1.895214166 Break this problem into two subproblems by setting (z + -0.857142857) equal to 1.895214166 and -1.895214166.Subproblem 1
z + -0.857142857 = 1.895214166 Simplifying z + -0.857142857 = 1.895214166 Reorder the terms: -0.857142857 + z = 1.895214166 Solving -0.857142857 + z = 1.895214166 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.857142857' to each side of the equation. -0.857142857 + 0.857142857 + z = 1.895214166 + 0.857142857 Combine like terms: -0.857142857 + 0.857142857 = 0.000000000 0.000000000 + z = 1.895214166 + 0.857142857 z = 1.895214166 + 0.857142857 Combine like terms: 1.895214166 + 0.857142857 = 2.752357023 z = 2.752357023 Simplifying z = 2.752357023Subproblem 2
z + -0.857142857 = -1.895214166 Simplifying z + -0.857142857 = -1.895214166 Reorder the terms: -0.857142857 + z = -1.895214166 Solving -0.857142857 + z = -1.895214166 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.857142857' to each side of the equation. -0.857142857 + 0.857142857 + z = -1.895214166 + 0.857142857 Combine like terms: -0.857142857 + 0.857142857 = 0.000000000 0.000000000 + z = -1.895214166 + 0.857142857 z = -1.895214166 + 0.857142857 Combine like terms: -1.895214166 + 0.857142857 = -1.038071309 z = -1.038071309 Simplifying z = -1.038071309Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.752357023, -1.038071309}
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