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Simplifying 3z2 + -7z + -3 = 0 Reorder the terms: -3 + -7z + 3z2 = 0 Solving -3 + -7z + 3z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1 + -2.333333333z + z2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -2.333333333z + 1 + z2 = 0 + 1 Reorder the terms: -1 + 1 + -2.333333333z + z2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -2.333333333z + z2 = 0 + 1 -2.333333333z + z2 = 0 + 1 Combine like terms: 0 + 1 = 1 -2.333333333z + z2 = 1 The z term is -2.333333333z. Take half its coefficient (-1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. -2.333333333z + 1.361111112 + z2 = 1 + 1.361111112 Reorder the terms: 1.361111112 + -2.333333333z + z2 = 1 + 1.361111112 Combine like terms: 1 + 1.361111112 = 2.361111112 1.361111112 + -2.333333333z + z2 = 2.361111112 Factor a perfect square on the left side: (z + -1.166666667)(z + -1.166666667) = 2.361111112 Calculate the square root of the right side: 1.536590743 Break this problem into two subproblems by setting (z + -1.166666667) equal to 1.536590743 and -1.536590743.Subproblem 1
z + -1.166666667 = 1.536590743 Simplifying z + -1.166666667 = 1.536590743 Reorder the terms: -1.166666667 + z = 1.536590743 Solving -1.166666667 + z = 1.536590743 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + z = 1.536590743 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + z = 1.536590743 + 1.166666667 z = 1.536590743 + 1.166666667 Combine like terms: 1.536590743 + 1.166666667 = 2.70325741 z = 2.70325741 Simplifying z = 2.70325741Subproblem 2
z + -1.166666667 = -1.536590743 Simplifying z + -1.166666667 = -1.536590743 Reorder the terms: -1.166666667 + z = -1.536590743 Solving -1.166666667 + z = -1.536590743 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + z = -1.536590743 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + z = -1.536590743 + 1.166666667 z = -1.536590743 + 1.166666667 Combine like terms: -1.536590743 + 1.166666667 = -0.369924076 z = -0.369924076 Simplifying z = -0.369924076Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.70325741, -0.369924076}
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