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

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


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

Reorder the terms:
-4z(-3 + z) = 2 + -3z
(-3 * -4z + z * -4z) = 2 + -3z
(12z + -4z2) = 2 + -3z

Solving
12z + -4z2 = 2 + -3z

Solving for variable 'z'.

Reorder the terms:
-2 + 12z + 3z + -4z2 = 2 + -3z + -2 + 3z

Combine like terms: 12z + 3z = 15z
-2 + 15z + -4z2 = 2 + -3z + -2 + 3z

Reorder the terms:
-2 + 15z + -4z2 = 2 + -2 + -3z + 3z

Combine like terms: 2 + -2 = 0
-2 + 15z + -4z2 = 0 + -3z + 3z
-2 + 15z + -4z2 = -3z + 3z

Combine like terms: -3z + 3z = 0
-2 + 15z + -4z2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
0.5 + -3.75z + z2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -3.75z + -0.5 + z2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -3.75z + z2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -3.75z + z2 = 0 + -0.5
-3.75z + z2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-3.75z + z2 = -0.5

The z term is -3.75z.  Take half its coefficient (-1.875).
Square it (3.515625) and add it to both sides.

Add '3.515625' to each side of the equation.
-3.75z + 3.515625 + z2 = -0.5 + 3.515625

Reorder the terms:
3.515625 + -3.75z + z2 = -0.5 + 3.515625

Combine like terms: -0.5 + 3.515625 = 3.015625
3.515625 + -3.75z + z2 = 3.015625

Factor a perfect square on the left side:
(z + -1.875)(z + -1.875) = 3.015625

Calculate the square root of the right side: 1.736555499

Break this problem into two subproblems by setting 
(z + -1.875) equal to 1.736555499 and -1.736555499.

Subproblem 1

z + -1.875 = 1.736555499 Simplifying z + -1.875 = 1.736555499 Reorder the terms: -1.875 + z = 1.736555499 Solving -1.875 + z = 1.736555499 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + z = 1.736555499 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + z = 1.736555499 + 1.875 z = 1.736555499 + 1.875 Combine like terms: 1.736555499 + 1.875 = 3.611555499 z = 3.611555499 Simplifying z = 3.611555499

Subproblem 2

z + -1.875 = -1.736555499 Simplifying z + -1.875 = -1.736555499 Reorder the terms: -1.875 + z = -1.736555499 Solving -1.875 + z = -1.736555499 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + z = -1.736555499 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + z = -1.736555499 + 1.875 z = -1.736555499 + 1.875 Combine like terms: -1.736555499 + 1.875 = 0.138444501 z = 0.138444501 Simplifying z = 0.138444501

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.611555499, 0.138444501}

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