(2z-3)(3z+4)=(6z-5)(4z+9)

Solution for (2z-3)(3z+4)=(6z-5)(4z+9) equation:

Simplifying
(2z + -3)(3z + 4) = (6z + -5)(4z + 9)

Reorder the terms:
(-3 + 2z)(3z + 4) = (6z + -5)(4z + 9)

Reorder the terms:
(-3 + 2z)(4 + 3z) = (6z + -5)(4z + 9)

Multiply (-3 + 2z) * (4 + 3z)
(-3(4 + 3z) + 2z * (4 + 3z)) = (6z + -5)(4z + 9)
((4 * -3 + 3z * -3) + 2z * (4 + 3z)) = (6z + -5)(4z + 9)
((-12 + -9z) + 2z * (4 + 3z)) = (6z + -5)(4z + 9)
(-12 + -9z + (4 * 2z + 3z * 2z)) = (6z + -5)(4z + 9)
(-12 + -9z + (8z + 6z2)) = (6z + -5)(4z + 9)

Combine like terms: -9z + 8z = -1z
(-12 + -1z + 6z2) = (6z + -5)(4z + 9)

Reorder the terms:
-12 + -1z + 6z2 = (-5 + 6z)(4z + 9)

Reorder the terms:
-12 + -1z + 6z2 = (-5 + 6z)(9 + 4z)

Multiply (-5 + 6z) * (9 + 4z)
-12 + -1z + 6z2 = (-5(9 + 4z) + 6z * (9 + 4z))
-12 + -1z + 6z2 = ((9 * -5 + 4z * -5) + 6z * (9 + 4z))
-12 + -1z + 6z2 = ((-45 + -20z) + 6z * (9 + 4z))
-12 + -1z + 6z2 = (-45 + -20z + (9 * 6z + 4z * 6z))
-12 + -1z + 6z2 = (-45 + -20z + (54z + 24z2))

Combine like terms: -20z + 54z = 34z
-12 + -1z + 6z2 = (-45 + 34z + 24z2)

Solving
-12 + -1z + 6z2 = -45 + 34z + 24z2

Solving for variable 'z'.

Reorder the terms:
-12 + 45 + -1z + -34z + 6z2 + -24z2 = -45 + 34z + 24z2 + 45 + -34z + -24z2

Combine like terms: -12 + 45 = 33
33 + -1z + -34z + 6z2 + -24z2 = -45 + 34z + 24z2 + 45 + -34z + -24z2

Combine like terms: -1z + -34z = -35z
33 + -35z + 6z2 + -24z2 = -45 + 34z + 24z2 + 45 + -34z + -24z2

Combine like terms: 6z2 + -24z2 = -18z2
33 + -35z + -18z2 = -45 + 34z + 24z2 + 45 + -34z + -24z2

Reorder the terms:
33 + -35z + -18z2 = -45 + 45 + 34z + -34z + 24z2 + -24z2

Combine like terms: -45 + 45 = 0
33 + -35z + -18z2 = 0 + 34z + -34z + 24z2 + -24z2
33 + -35z + -18z2 = 34z + -34z + 24z2 + -24z2

Combine like terms: 34z + -34z = 0
33 + -35z + -18z2 = 0 + 24z2 + -24z2
33 + -35z + -18z2 = 24z2 + -24z2

Combine like terms: 24z2 + -24z2 = 0
33 + -35z + -18z2 = 0

Begin completing the square.  Divide all terms by
-18 the coefficient of the squared term: 

Divide each side by '-18'.
-1.833333333 + 1.944444444z + z2 = 0

Move the constant term to the right:

Add '1.833333333' to each side of the equation.
-1.833333333 + 1.944444444z + 1.833333333 + z2 = 0 + 1.833333333

Reorder the terms:
-1.833333333 + 1.833333333 + 1.944444444z + z2 = 0 + 1.833333333

Combine like terms: -1.833333333 + 1.833333333 = 0.000000000
0.000000000 + 1.944444444z + z2 = 0 + 1.833333333
1.944444444z + z2 = 0 + 1.833333333

Combine like terms: 0 + 1.833333333 = 1.833333333
1.944444444z + z2 = 1.833333333

The z term is 1.944444444z.  Take half its coefficient (0.972222222).
Square it (0.9452160490) and add it to both sides.

Add '0.9452160490' to each side of the equation.
1.944444444z + 0.9452160490 + z2 = 1.833333333 + 0.9452160490

Reorder the terms:
0.9452160490 + 1.944444444z + z2 = 1.833333333 + 0.9452160490

Combine like terms: 1.833333333 + 0.9452160490 = 2.778549382
0.9452160490 + 1.944444444z + z2 = 2.778549382

Factor a perfect square on the left side:
(z + 0.972222222)(z + 0.972222222) = 2.778549382

Calculate the square root of the right side: 1.666898132

Break this problem into two subproblems by setting 
(z + 0.972222222) equal to 1.666898132 and -1.666898132.

Subproblem 1
z + 0.972222222 = 1.666898132

Simplifying
z + 0.972222222 = 1.666898132

Reorder the terms:
0.972222222 + z = 1.666898132

Solving
0.972222222 + z = 1.666898132

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-0.972222222' to each side of the equation.
0.972222222 + -0.972222222 + z = 1.666898132 + -0.972222222

Combine like terms: 0.972222222 + -0.972222222 = 0.000000000
0.000000000 + z = 1.666898132 + -0.972222222
z = 1.666898132 + -0.972222222

Combine like terms: 1.666898132 + -0.972222222 = 0.69467591
z = 0.69467591

Simplifying
z = 0.69467591

Subproblem 2
z + 0.972222222 = -1.666898132

Simplifying
z + 0.972222222 = -1.666898132

Reorder the terms:
0.972222222 + z = -1.666898132

Solving
0.972222222 + z = -1.666898132

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-0.972222222' to each side of the equation.
0.972222222 + -0.972222222 + z = -1.666898132 + -0.972222222

Combine like terms: 0.972222222 + -0.972222222 = 0.000000000
0.000000000 + z = -1.666898132 + -0.972222222
z = -1.666898132 + -0.972222222

Combine like terms: -1.666898132 + -0.972222222 = -2.639120354
z = -2.639120354

Simplifying
z = -2.639120354

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.69467591, -2.639120354}