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Simplifying 32z2 + 19z = 15 Reorder the terms: 19z + 32z2 = 15 Solving 19z + 32z2 = 15 Solving for variable 'z'. Reorder the terms: -15 + 19z + 32z2 = 15 + -15 Combine like terms: 15 + -15 = 0 -15 + 19z + 32z2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -0.46875 + 0.59375z + z2 = 0 Move the constant term to the right: Add '0.46875' to each side of the equation. -0.46875 + 0.59375z + 0.46875 + z2 = 0 + 0.46875 Reorder the terms: -0.46875 + 0.46875 + 0.59375z + z2 = 0 + 0.46875 Combine like terms: -0.46875 + 0.46875 = 0.00000 0.00000 + 0.59375z + z2 = 0 + 0.46875 0.59375z + z2 = 0 + 0.46875 Combine like terms: 0 + 0.46875 = 0.46875 0.59375z + z2 = 0.46875 The z term is 0.59375z. Take half its coefficient (0.296875). Square it (0.08813476563) and add it to both sides. Add '0.08813476563' to each side of the equation. 0.59375z + 0.08813476563 + z2 = 0.46875 + 0.08813476563 Reorder the terms: 0.08813476563 + 0.59375z + z2 = 0.46875 + 0.08813476563 Combine like terms: 0.46875 + 0.08813476563 = 0.55688476563 0.08813476563 + 0.59375z + z2 = 0.55688476563 Factor a perfect square on the left side: (z + 0.296875)(z + 0.296875) = 0.55688476563 Calculate the square root of the right side: 0.746247121 Break this problem into two subproblems by setting (z + 0.296875) equal to 0.746247121 and -0.746247121.Subproblem 1
z + 0.296875 = 0.746247121 Simplifying z + 0.296875 = 0.746247121 Reorder the terms: 0.296875 + z = 0.746247121 Solving 0.296875 + z = 0.746247121 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.296875' to each side of the equation. 0.296875 + -0.296875 + z = 0.746247121 + -0.296875 Combine like terms: 0.296875 + -0.296875 = 0.000000 0.000000 + z = 0.746247121 + -0.296875 z = 0.746247121 + -0.296875 Combine like terms: 0.746247121 + -0.296875 = 0.449372121 z = 0.449372121 Simplifying z = 0.449372121Subproblem 2
z + 0.296875 = -0.746247121 Simplifying z + 0.296875 = -0.746247121 Reorder the terms: 0.296875 + z = -0.746247121 Solving 0.296875 + z = -0.746247121 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.296875' to each side of the equation. 0.296875 + -0.296875 + z = -0.746247121 + -0.296875 Combine like terms: 0.296875 + -0.296875 = 0.000000 0.000000 + z = -0.746247121 + -0.296875 z = -0.746247121 + -0.296875 Combine like terms: -0.746247121 + -0.296875 = -1.043122121 z = -1.043122121 Simplifying z = -1.043122121Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.449372121, -1.043122121}
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