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Simplifying 2z2 + 15z + -3 = 13z Reorder the terms: -3 + 15z + 2z2 = 13z Solving -3 + 15z + 2z2 = 13z Solving for variable 'z'. Reorder the terms: -3 + 15z + -13z + 2z2 = 13z + -13z Combine like terms: 15z + -13z = 2z -3 + 2z + 2z2 = 13z + -13z Combine like terms: 13z + -13z = 0 -3 + 2z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + z + z2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + z + 1.5 + z2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + z + z2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z + z2 = 0 + 1.5 z + z2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 z + z2 = 1.5 The z term is z. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + z2 = 1.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + z2 = 1.5 + 0.25 Combine like terms: 1.5 + 0.25 = 1.75 1.25 + z2 = 1.75 Factor a perfect square on the left side: (z + 0.5)(z + 0.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (z + 0.5) equal to 1.322875656 and -1.322875656.Subproblem 1
z + 0.5 = 1.322875656 Simplifying z + 0.5 = 1.322875656 Reorder the terms: 0.5 + z = 1.322875656 Solving 0.5 + z = 1.322875656 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 1.322875656 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 1.322875656 + -0.5 z = 1.322875656 + -0.5 Combine like terms: 1.322875656 + -0.5 = 0.822875656 z = 0.822875656 Simplifying z = 0.822875656Subproblem 2
z + 0.5 = -1.322875656 Simplifying z + 0.5 = -1.322875656 Reorder the terms: 0.5 + z = -1.322875656 Solving 0.5 + z = -1.322875656 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -1.322875656 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -1.322875656 + -0.5 z = -1.322875656 + -0.5 Combine like terms: -1.322875656 + -0.5 = -1.822875656 z = -1.822875656 Simplifying z = -1.822875656Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.822875656, -1.822875656}
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