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Simplifying 22z + 22 = 7z2 + 33z + -48 Reorder the terms: 22 + 22z = 7z2 + 33z + -48 Reorder the terms: 22 + 22z = -48 + 33z + 7z2 Solving 22 + 22z = -48 + 33z + 7z2 Solving for variable 'z'. Reorder the terms: 22 + 48 + 22z + -33z + -7z2 = -48 + 33z + 7z2 + 48 + -33z + -7z2 Combine like terms: 22 + 48 = 70 70 + 22z + -33z + -7z2 = -48 + 33z + 7z2 + 48 + -33z + -7z2 Combine like terms: 22z + -33z = -11z 70 + -11z + -7z2 = -48 + 33z + 7z2 + 48 + -33z + -7z2 Reorder the terms: 70 + -11z + -7z2 = -48 + 48 + 33z + -33z + 7z2 + -7z2 Combine like terms: -48 + 48 = 0 70 + -11z + -7z2 = 0 + 33z + -33z + 7z2 + -7z2 70 + -11z + -7z2 = 33z + -33z + 7z2 + -7z2 Combine like terms: 33z + -33z = 0 70 + -11z + -7z2 = 0 + 7z2 + -7z2 70 + -11z + -7z2 = 7z2 + -7z2 Combine like terms: 7z2 + -7z2 = 0 70 + -11z + -7z2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -10 + 1.571428571z + z2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + 1.571428571z + 10 + z2 = 0 + 10 Reorder the terms: -10 + 10 + 1.571428571z + z2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + 1.571428571z + z2 = 0 + 10 1.571428571z + z2 = 0 + 10 Combine like terms: 0 + 10 = 10 1.571428571z + z2 = 10 The z term is 1.571428571z. Take half its coefficient (0.7857142855). Square it (0.6173469384) and add it to both sides. Add '0.6173469384' to each side of the equation. 1.571428571z + 0.6173469384 + z2 = 10 + 0.6173469384 Reorder the terms: 0.6173469384 + 1.571428571z + z2 = 10 + 0.6173469384 Combine like terms: 10 + 0.6173469384 = 10.6173469384 0.6173469384 + 1.571428571z + z2 = 10.6173469384 Factor a perfect square on the left side: (z + 0.7857142855)(z + 0.7857142855) = 10.6173469384 Calculate the square root of the right side: 3.258427065 Break this problem into two subproblems by setting (z + 0.7857142855) equal to 3.258427065 and -3.258427065.Subproblem 1
z + 0.7857142855 = 3.258427065 Simplifying z + 0.7857142855 = 3.258427065 Reorder the terms: 0.7857142855 + z = 3.258427065 Solving 0.7857142855 + z = 3.258427065 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.7857142855' to each side of the equation. 0.7857142855 + -0.7857142855 + z = 3.258427065 + -0.7857142855 Combine like terms: 0.7857142855 + -0.7857142855 = 0.0000000000 0.0000000000 + z = 3.258427065 + -0.7857142855 z = 3.258427065 + -0.7857142855 Combine like terms: 3.258427065 + -0.7857142855 = 2.4727127795 z = 2.4727127795 Simplifying z = 2.4727127795Subproblem 2
z + 0.7857142855 = -3.258427065 Simplifying z + 0.7857142855 = -3.258427065 Reorder the terms: 0.7857142855 + z = -3.258427065 Solving 0.7857142855 + z = -3.258427065 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.7857142855' to each side of the equation. 0.7857142855 + -0.7857142855 + z = -3.258427065 + -0.7857142855 Combine like terms: 0.7857142855 + -0.7857142855 = 0.0000000000 0.0000000000 + z = -3.258427065 + -0.7857142855 z = -3.258427065 + -0.7857142855 Combine like terms: -3.258427065 + -0.7857142855 = -4.0441413505 z = -4.0441413505 Simplifying z = -4.0441413505Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.4727127795, -4.0441413505}
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