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Simplifying (6z + 5)(z + -8) = 8 Reorder the terms: (5 + 6z)(z + -8) = 8 Reorder the terms: (5 + 6z)(-8 + z) = 8 Multiply (5 + 6z) * (-8 + z) (5(-8 + z) + 6z * (-8 + z)) = 8 ((-8 * 5 + z * 5) + 6z * (-8 + z)) = 8 ((-40 + 5z) + 6z * (-8 + z)) = 8 (-40 + 5z + (-8 * 6z + z * 6z)) = 8 (-40 + 5z + (-48z + 6z2)) = 8 Combine like terms: 5z + -48z = -43z (-40 + -43z + 6z2) = 8 Solving -40 + -43z + 6z2 = 8 Solving for variable 'z'. Reorder the terms: -40 + -8 + -43z + 6z2 = 8 + -8 Combine like terms: -40 + -8 = -48 -48 + -43z + 6z2 = 8 + -8 Combine like terms: 8 + -8 = 0 -48 + -43z + 6z2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -8 + -7.166666667z + z2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + -7.166666667z + 8 + z2 = 0 + 8 Reorder the terms: -8 + 8 + -7.166666667z + z2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + -7.166666667z + z2 = 0 + 8 -7.166666667z + z2 = 0 + 8 Combine like terms: 0 + 8 = 8 -7.166666667z + z2 = 8 The z term is -7.166666667z. Take half its coefficient (-3.583333334). Square it (12.84027778) and add it to both sides. Add '12.84027778' to each side of the equation. -7.166666667z + 12.84027778 + z2 = 8 + 12.84027778 Reorder the terms: 12.84027778 + -7.166666667z + z2 = 8 + 12.84027778 Combine like terms: 8 + 12.84027778 = 20.84027778 12.84027778 + -7.166666667z + z2 = 20.84027778 Factor a perfect square on the left side: (z + -3.583333334)(z + -3.583333334) = 20.84027778 Calculate the square root of the right side: 4.565115309 Break this problem into two subproblems by setting (z + -3.583333334) equal to 4.565115309 and -4.565115309.Subproblem 1
z + -3.583333334 = 4.565115309 Simplifying z + -3.583333334 = 4.565115309 Reorder the terms: -3.583333334 + z = 4.565115309 Solving -3.583333334 + z = 4.565115309 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3.583333334' to each side of the equation. -3.583333334 + 3.583333334 + z = 4.565115309 + 3.583333334 Combine like terms: -3.583333334 + 3.583333334 = 0.000000000 0.000000000 + z = 4.565115309 + 3.583333334 z = 4.565115309 + 3.583333334 Combine like terms: 4.565115309 + 3.583333334 = 8.148448643 z = 8.148448643 Simplifying z = 8.148448643Subproblem 2
z + -3.583333334 = -4.565115309 Simplifying z + -3.583333334 = -4.565115309 Reorder the terms: -3.583333334 + z = -4.565115309 Solving -3.583333334 + z = -4.565115309 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3.583333334' to each side of the equation. -3.583333334 + 3.583333334 + z = -4.565115309 + 3.583333334 Combine like terms: -3.583333334 + 3.583333334 = 0.000000000 0.000000000 + z = -4.565115309 + 3.583333334 z = -4.565115309 + 3.583333334 Combine like terms: -4.565115309 + 3.583333334 = -0.981781975 z = -0.981781975 Simplifying z = -0.981781975Solution
The solution to the problem is based on the solutions from the subproblems. z = {8.148448643, -0.981781975}
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