-2(3-4z)+7z=12z(z+4)

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Solution for -2(3-4z)+7z=12z(z+4) equation:


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 = -0.195752358

Subproblem 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 = -2.554247642

Solution

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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