(6z-5)(3z-4)=180

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Solution for (6z-5)(3z-4)=180 equation:


Simplifying
(6z + -5)(3z + -4) = 180

Reorder the terms:
(-5 + 6z)(3z + -4) = 180

Reorder the terms:
(-5 + 6z)(-4 + 3z) = 180

Multiply (-5 + 6z) * (-4 + 3z)
(-5(-4 + 3z) + 6z * (-4 + 3z)) = 180
((-4 * -5 + 3z * -5) + 6z * (-4 + 3z)) = 180
((20 + -15z) + 6z * (-4 + 3z)) = 180
(20 + -15z + (-4 * 6z + 3z * 6z)) = 180
(20 + -15z + (-24z + 18z2)) = 180

Combine like terms: -15z + -24z = -39z
(20 + -39z + 18z2) = 180

Solving
20 + -39z + 18z2 = 180

Solving for variable 'z'.

Reorder the terms:
20 + -180 + -39z + 18z2 = 180 + -180

Combine like terms: 20 + -180 = -160
-160 + -39z + 18z2 = 180 + -180

Combine like terms: 180 + -180 = 0
-160 + -39z + 18z2 = 0

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

Divide each side by '18'.
-8.888888889 + -2.166666667z + z2 = 0

Move the constant term to the right:

Add '8.888888889' to each side of the equation.
-8.888888889 + -2.166666667z + 8.888888889 + z2 = 0 + 8.888888889

Reorder the terms:
-8.888888889 + 8.888888889 + -2.166666667z + z2 = 0 + 8.888888889

Combine like terms: -8.888888889 + 8.888888889 = 0.000000000
0.000000000 + -2.166666667z + z2 = 0 + 8.888888889
-2.166666667z + z2 = 0 + 8.888888889

Combine like terms: 0 + 8.888888889 = 8.888888889
-2.166666667z + z2 = 8.888888889

The z term is -2.166666667z.  Take half its coefficient (-1.083333334).
Square it (1.173611113) and add it to both sides.

Add '1.173611113' to each side of the equation.
-2.166666667z + 1.173611113 + z2 = 8.888888889 + 1.173611113

Reorder the terms:
1.173611113 + -2.166666667z + z2 = 8.888888889 + 1.173611113

Combine like terms: 8.888888889 + 1.173611113 = 10.062500002
1.173611113 + -2.166666667z + z2 = 10.062500002

Factor a perfect square on the left side:
(z + -1.083333334)(z + -1.083333334) = 10.062500002

Calculate the square root of the right side: 3.172144385

Break this problem into two subproblems by setting 
(z + -1.083333334) equal to 3.172144385 and -3.172144385.

Subproblem 1

z + -1.083333334 = 3.172144385 Simplifying z + -1.083333334 = 3.172144385 Reorder the terms: -1.083333334 + z = 3.172144385 Solving -1.083333334 + z = 3.172144385 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + z = 3.172144385 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + z = 3.172144385 + 1.083333334 z = 3.172144385 + 1.083333334 Combine like terms: 3.172144385 + 1.083333334 = 4.255477719 z = 4.255477719 Simplifying z = 4.255477719

Subproblem 2

z + -1.083333334 = -3.172144385 Simplifying z + -1.083333334 = -3.172144385 Reorder the terms: -1.083333334 + z = -3.172144385 Solving -1.083333334 + z = -3.172144385 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + z = -3.172144385 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + z = -3.172144385 + 1.083333334 z = -3.172144385 + 1.083333334 Combine like terms: -3.172144385 + 1.083333334 = -2.088811051 z = -2.088811051 Simplifying z = -2.088811051

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.255477719, -2.088811051}

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