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Simplifying 6z2 + -9 = 4z Reorder the terms: -9 + 6z2 = 4z Solving -9 + 6z2 = 4z Solving for variable 'z'. Reorder the terms: -9 + -4z + 6z2 = 4z + -4z Combine like terms: 4z + -4z = 0 -9 + -4z + 6z2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.5 + -0.6666666667z + z2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -0.6666666667z + 1.5 + z2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -0.6666666667z + z2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -0.6666666667z + z2 = 0 + 1.5 -0.6666666667z + z2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -0.6666666667z + z2 = 1.5 The z term is -0.6666666667z. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667z + 0.1111111112 + z2 = 1.5 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667z + z2 = 1.5 + 0.1111111112 Combine like terms: 1.5 + 0.1111111112 = 1.6111111112 0.1111111112 + -0.6666666667z + z2 = 1.6111111112 Factor a perfect square on the left side: (z + -0.3333333334)(z + -0.3333333334) = 1.6111111112 Calculate the square root of the right side: 1.269295518 Break this problem into two subproblems by setting (z + -0.3333333334) equal to 1.269295518 and -1.269295518.Subproblem 1
z + -0.3333333334 = 1.269295518 Simplifying z + -0.3333333334 = 1.269295518 Reorder the terms: -0.3333333334 + z = 1.269295518 Solving -0.3333333334 + z = 1.269295518 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + z = 1.269295518 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + z = 1.269295518 + 0.3333333334 z = 1.269295518 + 0.3333333334 Combine like terms: 1.269295518 + 0.3333333334 = 1.6026288514 z = 1.6026288514 Simplifying z = 1.6026288514Subproblem 2
z + -0.3333333334 = -1.269295518 Simplifying z + -0.3333333334 = -1.269295518 Reorder the terms: -0.3333333334 + z = -1.269295518 Solving -0.3333333334 + z = -1.269295518 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + z = -1.269295518 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + z = -1.269295518 + 0.3333333334 z = -1.269295518 + 0.3333333334 Combine like terms: -1.269295518 + 0.3333333334 = -0.9359621846 z = -0.9359621846 Simplifying z = -0.9359621846Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.6026288514, -0.9359621846}
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