(2z+6)(4z-3)=(5z+12)

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Solution for (2z+6)(4z-3)=(5z+12) equation:


Simplifying
(2z + 6)(4z + -3) = (5z + 12)

Reorder the terms:
(6 + 2z)(4z + -3) = (5z + 12)

Reorder the terms:
(6 + 2z)(-3 + 4z) = (5z + 12)

Multiply (6 + 2z) * (-3 + 4z)
(6(-3 + 4z) + 2z * (-3 + 4z)) = (5z + 12)
((-3 * 6 + 4z * 6) + 2z * (-3 + 4z)) = (5z + 12)
((-18 + 24z) + 2z * (-3 + 4z)) = (5z + 12)
(-18 + 24z + (-3 * 2z + 4z * 2z)) = (5z + 12)
(-18 + 24z + (-6z + 8z2)) = (5z + 12)

Combine like terms: 24z + -6z = 18z
(-18 + 18z + 8z2) = (5z + 12)

Reorder the terms:
-18 + 18z + 8z2 = (12 + 5z)

Remove parenthesis around (12 + 5z)
-18 + 18z + 8z2 = 12 + 5z

Solving
-18 + 18z + 8z2 = 12 + 5z

Solving for variable 'z'.

Reorder the terms:
-18 + -12 + 18z + -5z + 8z2 = 12 + 5z + -12 + -5z

Combine like terms: -18 + -12 = -30
-30 + 18z + -5z + 8z2 = 12 + 5z + -12 + -5z

Combine like terms: 18z + -5z = 13z
-30 + 13z + 8z2 = 12 + 5z + -12 + -5z

Reorder the terms:
-30 + 13z + 8z2 = 12 + -12 + 5z + -5z

Combine like terms: 12 + -12 = 0
-30 + 13z + 8z2 = 0 + 5z + -5z
-30 + 13z + 8z2 = 5z + -5z

Combine like terms: 5z + -5z = 0
-30 + 13z + 8z2 = 0

Begin completing the square.  Divide all terms by
8 the coefficient of the squared term: 

Divide each side by '8'.
-3.75 + 1.625z + z2 = 0

Move the constant term to the right:

Add '3.75' to each side of the equation.
-3.75 + 1.625z + 3.75 + z2 = 0 + 3.75

Reorder the terms:
-3.75 + 3.75 + 1.625z + z2 = 0 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + 1.625z + z2 = 0 + 3.75
1.625z + z2 = 0 + 3.75

Combine like terms: 0 + 3.75 = 3.75
1.625z + z2 = 3.75

The z term is 1.625z.  Take half its coefficient (0.8125).
Square it (0.66015625) and add it to both sides.

Add '0.66015625' to each side of the equation.
1.625z + 0.66015625 + z2 = 3.75 + 0.66015625

Reorder the terms:
0.66015625 + 1.625z + z2 = 3.75 + 0.66015625

Combine like terms: 3.75 + 0.66015625 = 4.41015625
0.66015625 + 1.625z + z2 = 4.41015625

Factor a perfect square on the left side:
(z + 0.8125)(z + 0.8125) = 4.41015625

Calculate the square root of the right side: 2.100037202

Break this problem into two subproblems by setting 
(z + 0.8125) equal to 2.100037202 and -2.100037202.

Subproblem 1

z + 0.8125 = 2.100037202 Simplifying z + 0.8125 = 2.100037202 Reorder the terms: 0.8125 + z = 2.100037202 Solving 0.8125 + z = 2.100037202 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.8125' to each side of the equation. 0.8125 + -0.8125 + z = 2.100037202 + -0.8125 Combine like terms: 0.8125 + -0.8125 = 0.0000 0.0000 + z = 2.100037202 + -0.8125 z = 2.100037202 + -0.8125 Combine like terms: 2.100037202 + -0.8125 = 1.287537202 z = 1.287537202 Simplifying z = 1.287537202

Subproblem 2

z + 0.8125 = -2.100037202 Simplifying z + 0.8125 = -2.100037202 Reorder the terms: 0.8125 + z = -2.100037202 Solving 0.8125 + z = -2.100037202 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.8125' to each side of the equation. 0.8125 + -0.8125 + z = -2.100037202 + -0.8125 Combine like terms: 0.8125 + -0.8125 = 0.0000 0.0000 + z = -2.100037202 + -0.8125 z = -2.100037202 + -0.8125 Combine like terms: -2.100037202 + -0.8125 = -2.912537202 z = -2.912537202 Simplifying z = -2.912537202

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.287537202, -2.912537202}

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