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Simplifying 3z2 + 13z = 21 Reorder the terms: 13z + 3z2 = 21 Solving 13z + 3z2 = 21 Solving for variable 'z'. Reorder the terms: -21 + 13z + 3z2 = 21 + -21 Combine like terms: 21 + -21 = 0 -21 + 13z + 3z2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -7 + 4.333333333z + z2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + 4.333333333z + 7 + z2 = 0 + 7 Reorder the terms: -7 + 7 + 4.333333333z + z2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 4.333333333z + z2 = 0 + 7 4.333333333z + z2 = 0 + 7 Combine like terms: 0 + 7 = 7 4.333333333z + z2 = 7 The z term is 4.333333333z. Take half its coefficient (2.166666667). Square it (4.694444446) and add it to both sides. Add '4.694444446' to each side of the equation. 4.333333333z + 4.694444446 + z2 = 7 + 4.694444446 Reorder the terms: 4.694444446 + 4.333333333z + z2 = 7 + 4.694444446 Combine like terms: 7 + 4.694444446 = 11.694444446 4.694444446 + 4.333333333z + z2 = 11.694444446 Factor a perfect square on the left side: (z + 2.166666667)(z + 2.166666667) = 11.694444446 Calculate the square root of the right side: 3.419714088 Break this problem into two subproblems by setting (z + 2.166666667) equal to 3.419714088 and -3.419714088.Subproblem 1
z + 2.166666667 = 3.419714088 Simplifying z + 2.166666667 = 3.419714088 Reorder the terms: 2.166666667 + z = 3.419714088 Solving 2.166666667 + z = 3.419714088 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + z = 3.419714088 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + z = 3.419714088 + -2.166666667 z = 3.419714088 + -2.166666667 Combine like terms: 3.419714088 + -2.166666667 = 1.253047421 z = 1.253047421 Simplifying z = 1.253047421Subproblem 2
z + 2.166666667 = -3.419714088 Simplifying z + 2.166666667 = -3.419714088 Reorder the terms: 2.166666667 + z = -3.419714088 Solving 2.166666667 + z = -3.419714088 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + z = -3.419714088 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + z = -3.419714088 + -2.166666667 z = -3.419714088 + -2.166666667 Combine like terms: -3.419714088 + -2.166666667 = -5.586380755 z = -5.586380755 Simplifying z = -5.586380755Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.253047421, -5.586380755}
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