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Simplifying 0 = 3z2 + -10z + -21 Reorder the terms: 0 = -21 + -10z + 3z2 Solving 0 = -21 + -10z + 3z2 Solving for variable 'z'. Combine like terms: 0 + 21 = 21 21 + 10z + -3z2 = -21 + -10z + 3z2 + 21 + 10z + -3z2 Reorder the terms: 21 + 10z + -3z2 = -21 + 21 + -10z + 10z + 3z2 + -3z2 Combine like terms: -21 + 21 = 0 21 + 10z + -3z2 = 0 + -10z + 10z + 3z2 + -3z2 21 + 10z + -3z2 = -10z + 10z + 3z2 + -3z2 Combine like terms: -10z + 10z = 0 21 + 10z + -3z2 = 0 + 3z2 + -3z2 21 + 10z + -3z2 = 3z2 + -3z2 Combine like terms: 3z2 + -3z2 = 0 21 + 10z + -3z2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -7 + -3.333333333z + z2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + -3.333333333z + 7 + z2 = 0 + 7 Reorder the terms: -7 + 7 + -3.333333333z + z2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + -3.333333333z + z2 = 0 + 7 -3.333333333z + z2 = 0 + 7 Combine like terms: 0 + 7 = 7 -3.333333333z + z2 = 7 The z term is -3.333333333z. Take half its coefficient (-1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. -3.333333333z + 2.777777779 + z2 = 7 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333z + z2 = 7 + 2.777777779 Combine like terms: 7 + 2.777777779 = 9.777777779 2.777777779 + -3.333333333z + z2 = 9.777777779 Factor a perfect square on the left side: (z + -1.666666667)(z + -1.666666667) = 9.777777779 Calculate the square root of the right side: 3.12694384 Break this problem into two subproblems by setting (z + -1.666666667) equal to 3.12694384 and -3.12694384.Subproblem 1
z + -1.666666667 = 3.12694384 Simplifying z + -1.666666667 = 3.12694384 Reorder the terms: -1.666666667 + z = 3.12694384 Solving -1.666666667 + z = 3.12694384 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + z = 3.12694384 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + z = 3.12694384 + 1.666666667 z = 3.12694384 + 1.666666667 Combine like terms: 3.12694384 + 1.666666667 = 4.793610507 z = 4.793610507 Simplifying z = 4.793610507Subproblem 2
z + -1.666666667 = -3.12694384 Simplifying z + -1.666666667 = -3.12694384 Reorder the terms: -1.666666667 + z = -3.12694384 Solving -1.666666667 + z = -3.12694384 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + z = -3.12694384 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + z = -3.12694384 + 1.666666667 z = -3.12694384 + 1.666666667 Combine like terms: -3.12694384 + 1.666666667 = -1.460277173 z = -1.460277173 Simplifying z = -1.460277173Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.793610507, -1.460277173}
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