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Simplifying 2z2 = 3z + 15 Reorder the terms: 2z2 = 15 + 3z Solving 2z2 = 15 + 3z Solving for variable 'z'. Reorder the terms: -15 + -3z + 2z2 = 15 + 3z + -15 + -3z Reorder the terms: -15 + -3z + 2z2 = 15 + -15 + 3z + -3z Combine like terms: 15 + -15 = 0 -15 + -3z + 2z2 = 0 + 3z + -3z -15 + -3z + 2z2 = 3z + -3z Combine like terms: 3z + -3z = 0 -15 + -3z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7.5 + -1.5z + z2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + -1.5z + 7.5 + z2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + -1.5z + z2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + -1.5z + z2 = 0 + 7.5 -1.5z + z2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 -1.5z + z2 = 7.5 The z term is -1.5z. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5z + 0.5625 + z2 = 7.5 + 0.5625 Reorder the terms: 0.5625 + -1.5z + z2 = 7.5 + 0.5625 Combine like terms: 7.5 + 0.5625 = 8.0625 0.5625 + -1.5z + z2 = 8.0625 Factor a perfect square on the left side: (z + -0.75)(z + -0.75) = 8.0625 Calculate the square root of the right side: 2.839454173 Break this problem into two subproblems by setting (z + -0.75) equal to 2.839454173 and -2.839454173.Subproblem 1
z + -0.75 = 2.839454173 Simplifying z + -0.75 = 2.839454173 Reorder the terms: -0.75 + z = 2.839454173 Solving -0.75 + z = 2.839454173 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = 2.839454173 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = 2.839454173 + 0.75 z = 2.839454173 + 0.75 Combine like terms: 2.839454173 + 0.75 = 3.589454173 z = 3.589454173 Simplifying z = 3.589454173Subproblem 2
z + -0.75 = -2.839454173 Simplifying z + -0.75 = -2.839454173 Reorder the terms: -0.75 + z = -2.839454173 Solving -0.75 + z = -2.839454173 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = -2.839454173 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = -2.839454173 + 0.75 z = -2.839454173 + 0.75 Combine like terms: -2.839454173 + 0.75 = -2.089454173 z = -2.089454173 Simplifying z = -2.089454173Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.589454173, -2.089454173}
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