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Simplifying 3z2 + 45 = 28z Reorder the terms: 45 + 3z2 = 28z Solving 45 + 3z2 = 28z Solving for variable 'z'. Reorder the terms: 45 + -28z + 3z2 = 28z + -28z Combine like terms: 28z + -28z = 0 45 + -28z + 3z2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 15 + -9.333333333z + z2 = 0 Move the constant term to the right: Add '-15' to each side of the equation. 15 + -9.333333333z + -15 + z2 = 0 + -15 Reorder the terms: 15 + -15 + -9.333333333z + z2 = 0 + -15 Combine like terms: 15 + -15 = 0 0 + -9.333333333z + z2 = 0 + -15 -9.333333333z + z2 = 0 + -15 Combine like terms: 0 + -15 = -15 -9.333333333z + z2 = -15 The z term is -9.333333333z. Take half its coefficient (-4.666666667). Square it (21.77777778) and add it to both sides. Add '21.77777778' to each side of the equation. -9.333333333z + 21.77777778 + z2 = -15 + 21.77777778 Reorder the terms: 21.77777778 + -9.333333333z + z2 = -15 + 21.77777778 Combine like terms: -15 + 21.77777778 = 6.77777778 21.77777778 + -9.333333333z + z2 = 6.77777778 Factor a perfect square on the left side: (z + -4.666666667)(z + -4.666666667) = 6.77777778 Calculate the square root of the right side: 2.603416559 Break this problem into two subproblems by setting (z + -4.666666667) equal to 2.603416559 and -2.603416559.Subproblem 1
z + -4.666666667 = 2.603416559 Simplifying z + -4.666666667 = 2.603416559 Reorder the terms: -4.666666667 + z = 2.603416559 Solving -4.666666667 + z = 2.603416559 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.666666667' to each side of the equation. -4.666666667 + 4.666666667 + z = 2.603416559 + 4.666666667 Combine like terms: -4.666666667 + 4.666666667 = 0.000000000 0.000000000 + z = 2.603416559 + 4.666666667 z = 2.603416559 + 4.666666667 Combine like terms: 2.603416559 + 4.666666667 = 7.270083226 z = 7.270083226 Simplifying z = 7.270083226Subproblem 2
z + -4.666666667 = -2.603416559 Simplifying z + -4.666666667 = -2.603416559 Reorder the terms: -4.666666667 + z = -2.603416559 Solving -4.666666667 + z = -2.603416559 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.666666667' to each side of the equation. -4.666666667 + 4.666666667 + z = -2.603416559 + 4.666666667 Combine like terms: -4.666666667 + 4.666666667 = 0.000000000 0.000000000 + z = -2.603416559 + 4.666666667 z = -2.603416559 + 4.666666667 Combine like terms: -2.603416559 + 4.666666667 = 2.063250108 z = 2.063250108 Simplifying z = 2.063250108Solution
The solution to the problem is based on the solutions from the subproblems. z = {7.270083226, 2.063250108}
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