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Simplifying -2(3 + -4z) + 7z = 12z(z + 4) (3 * -2 + -4z * -2) + 7z = 12z(z + 4) (-6 + 8z) + 7z = 12z(z + 4) Combine like terms: 8z + 7z = 15z -6 + 15z = 12z(z + 4) Reorder the terms: -6 + 15z = 12z(4 + z) -6 + 15z = (4 * 12z + z * 12z) -6 + 15z = (48z + 12z2) Solving -6 + 15z = 48z + 12z2 Solving for variable 'z'. Combine like terms: 15z + -48z = -33z -6 + -33z + -12z2 = 48z + 12z2 + -48z + -12z2 Reorder the terms: -6 + -33z + -12z2 = 48z + -48z + 12z2 + -12z2 Combine like terms: 48z + -48z = 0 -6 + -33z + -12z2 = 0 + 12z2 + -12z2 -6 + -33z + -12z2 = 12z2 + -12z2 Combine like terms: 12z2 + -12z2 = 0 -6 + -33z + -12z2 = 0 Factor out the Greatest Common Factor (GCF), '-3'. -3(2 + 11z + 4z2) = 0 Ignore the factor -3.Subproblem 1
Set the factor '(2 + 11z + 4z2)' equal to zero and attempt to solve: Simplifying 2 + 11z + 4z2 = 0 Solving 2 + 11z + 4z2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.5 + 2.75z + z2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 2.75z + -0.5 + z2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 2.75z + z2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 2.75z + z2 = 0 + -0.5 2.75z + z2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 2.75z + z2 = -0.5 The z term is 2.75z. Take half its coefficient (1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. 2.75z + 1.890625 + z2 = -0.5 + 1.890625 Reorder the terms: 1.890625 + 2.75z + z2 = -0.5 + 1.890625 Combine like terms: -0.5 + 1.890625 = 1.390625 1.890625 + 2.75z + z2 = 1.390625 Factor a perfect square on the left side: (z + 1.375)(z + 1.375) = 1.390625 Calculate the square root of the right side: 1.179247642 Break this problem into two subproblems by setting (z + 1.375) equal to 1.179247642 and -1.179247642.Subproblem 1
z + 1.375 = 1.179247642 Simplifying z + 1.375 = 1.179247642 Reorder the terms: 1.375 + z = 1.179247642 Solving 1.375 + z = 1.179247642 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + z = 1.179247642 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + z = 1.179247642 + -1.375 z = 1.179247642 + -1.375 Combine like terms: 1.179247642 + -1.375 = -0.195752358 z = -0.195752358 Simplifying z = -0.195752358Subproblem 2
z + 1.375 = -1.179247642 Simplifying z + 1.375 = -1.179247642 Reorder the terms: 1.375 + z = -1.179247642 Solving 1.375 + z = -1.179247642 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + z = -1.179247642 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + z = -1.179247642 + -1.375 z = -1.179247642 + -1.375 Combine like terms: -1.179247642 + -1.375 = -2.554247642 z = -2.554247642 Simplifying z = -2.554247642Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.195752358, -2.554247642}Solution
z = {-0.195752358, -2.554247642}
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