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Simplifying 12z2 + 15z + -2 = 0 Reorder the terms: -2 + 15z + 12z2 = 0 Solving -2 + 15z + 12z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.1666666667 + 1.25z + z2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + 1.25z + 0.1666666667 + z2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + 1.25z + z2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + 1.25z + z2 = 0 + 0.1666666667 1.25z + z2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 1.25z + z2 = 0.1666666667 The z term is 1.25z. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25z + 0.390625 + z2 = 0.1666666667 + 0.390625 Reorder the terms: 0.390625 + 1.25z + z2 = 0.1666666667 + 0.390625 Combine like terms: 0.1666666667 + 0.390625 = 0.5572916667 0.390625 + 1.25z + z2 = 0.5572916667 Factor a perfect square on the left side: (z + 0.625)(z + 0.625) = 0.5572916667 Calculate the square root of the right side: 0.746519703 Break this problem into two subproblems by setting (z + 0.625) equal to 0.746519703 and -0.746519703.Subproblem 1
z + 0.625 = 0.746519703 Simplifying z + 0.625 = 0.746519703 Reorder the terms: 0.625 + z = 0.746519703 Solving 0.625 + z = 0.746519703 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + z = 0.746519703 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + z = 0.746519703 + -0.625 z = 0.746519703 + -0.625 Combine like terms: 0.746519703 + -0.625 = 0.121519703 z = 0.121519703 Simplifying z = 0.121519703Subproblem 2
z + 0.625 = -0.746519703 Simplifying z + 0.625 = -0.746519703 Reorder the terms: 0.625 + z = -0.746519703 Solving 0.625 + z = -0.746519703 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + z = -0.746519703 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + z = -0.746519703 + -0.625 z = -0.746519703 + -0.625 Combine like terms: -0.746519703 + -0.625 = -1.371519703 z = -1.371519703 Simplifying z = -1.371519703Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.121519703, -1.371519703}
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