(3z-1)(z-4)=(2z+3)(z-3)-(z+1)

Solution for (3z-1)(z-4)=(2z+3)(z-3)-(z+1) equation:

Simplifying
(3z + -1)(z + -4) = (2z + 3)(z + -3) + -1(z + 1)

Reorder the terms:
(-1 + 3z)(z + -4) = (2z + 3)(z + -3) + -1(z + 1)

Reorder the terms:
(-1 + 3z)(-4 + z) = (2z + 3)(z + -3) + -1(z + 1)

Multiply (-1 + 3z) * (-4 + z)
(-1(-4 + z) + 3z * (-4 + z)) = (2z + 3)(z + -3) + -1(z + 1)
((-4 * -1 + z * -1) + 3z * (-4 + z)) = (2z + 3)(z + -3) + -1(z + 1)
((4 + -1z) + 3z * (-4 + z)) = (2z + 3)(z + -3) + -1(z + 1)
(4 + -1z + (-4 * 3z + z * 3z)) = (2z + 3)(z + -3) + -1(z + 1)
(4 + -1z + (-12z + 3z2)) = (2z + 3)(z + -3) + -1(z + 1)

Combine like terms: -1z + -12z = -13z
(4 + -13z + 3z2) = (2z + 3)(z + -3) + -1(z + 1)

Reorder the terms:
4 + -13z + 3z2 = (3 + 2z)(z + -3) + -1(z + 1)

Reorder the terms:
4 + -13z + 3z2 = (3 + 2z)(-3 + z) + -1(z + 1)

Multiply (3 + 2z) * (-3 + z)
4 + -13z + 3z2 = (3(-3 + z) + 2z * (-3 + z)) + -1(z + 1)
4 + -13z + 3z2 = ((-3 * 3 + z * 3) + 2z * (-3 + z)) + -1(z + 1)
4 + -13z + 3z2 = ((-9 + 3z) + 2z * (-3 + z)) + -1(z + 1)
4 + -13z + 3z2 = (-9 + 3z + (-3 * 2z + z * 2z)) + -1(z + 1)
4 + -13z + 3z2 = (-9 + 3z + (-6z + 2z2)) + -1(z + 1)

Combine like terms: 3z + -6z = -3z
4 + -13z + 3z2 = (-9 + -3z + 2z2) + -1(z + 1)

Reorder the terms:
4 + -13z + 3z2 = -9 + -3z + 2z2 + -1(1 + z)
4 + -13z + 3z2 = -9 + -3z + 2z2 + (1 * -1 + z * -1)
4 + -13z + 3z2 = -9 + -3z + 2z2 + (-1 + -1z)

Reorder the terms:
4 + -13z + 3z2 = -9 + -1 + -3z + -1z + 2z2

Combine like terms: -9 + -1 = -10
4 + -13z + 3z2 = -10 + -3z + -1z + 2z2

Combine like terms: -3z + -1z = -4z
4 + -13z + 3z2 = -10 + -4z + 2z2

Solving
4 + -13z + 3z2 = -10 + -4z + 2z2

Solving for variable 'z'.

Reorder the terms:
4 + 10 + -13z + 4z + 3z2 + -2z2 = -10 + -4z + 2z2 + 10 + 4z + -2z2

Combine like terms: 4 + 10 = 14
14 + -13z + 4z + 3z2 + -2z2 = -10 + -4z + 2z2 + 10 + 4z + -2z2

Combine like terms: -13z + 4z = -9z
14 + -9z + 3z2 + -2z2 = -10 + -4z + 2z2 + 10 + 4z + -2z2

Combine like terms: 3z2 + -2z2 = 1z2
14 + -9z + 1z2 = -10 + -4z + 2z2 + 10 + 4z + -2z2

Reorder the terms:
14 + -9z + 1z2 = -10 + 10 + -4z + 4z + 2z2 + -2z2

Combine like terms: -10 + 10 = 0
14 + -9z + 1z2 = 0 + -4z + 4z + 2z2 + -2z2
14 + -9z + 1z2 = -4z + 4z + 2z2 + -2z2

Combine like terms: -4z + 4z = 0
14 + -9z + 1z2 = 0 + 2z2 + -2z2
14 + -9z + 1z2 = 2z2 + -2z2

Combine like terms: 2z2 + -2z2 = 0
14 + -9z + 1z2 = 0

Factor a trinomial.
(2 + -1z)(7 + -1z) = 0

Subproblem 1
Set the factor '(2 + -1z)' equal to zero and attempt to solve:

Simplifying
2 + -1z = 0

Solving
2 + -1z = 0

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

Add '-2' to each side of the equation.
2 + -2 + -1z = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -1z = 0 + -2
-1z = 0 + -2

Combine like terms: 0 + -2 = -2
-1z = -2

Divide each side by '-1'.
z = 2

Simplifying
z = 2
Subproblem 2
Set the factor '(7 + -1z)' equal to zero and attempt to solve:

Simplifying
7 + -1z = 0

Solving
7 + -1z = 0

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

Add '-7' to each side of the equation.
7 + -7 + -1z = 0 + -7

Combine like terms: 7 + -7 = 0
0 + -1z = 0 + -7
-1z = 0 + -7

Combine like terms: 0 + -7 = -7
-1z = -7

Divide each side by '-1'.
z = 7

Simplifying
z = 7
Solution
z = {2, 7}