0=3z^2+2z-45

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Solution for 0=3z^2+2z-45 equation:


Simplifying
0 = 3z2 + 2z + -45

Reorder the terms:
0 = -45 + 2z + 3z2

Solving
0 = -45 + 2z + 3z2

Solving for variable 'z'.

Combine like terms: 0 + 45 = 45
45 + -2z + -3z2 = -45 + 2z + 3z2 + 45 + -2z + -3z2

Reorder the terms:
45 + -2z + -3z2 = -45 + 45 + 2z + -2z + 3z2 + -3z2

Combine like terms: -45 + 45 = 0
45 + -2z + -3z2 = 0 + 2z + -2z + 3z2 + -3z2
45 + -2z + -3z2 = 2z + -2z + 3z2 + -3z2

Combine like terms: 2z + -2z = 0
45 + -2z + -3z2 = 0 + 3z2 + -3z2
45 + -2z + -3z2 = 3z2 + -3z2

Combine like terms: 3z2 + -3z2 = 0
45 + -2z + -3z2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-15 + 0.6666666667z + z2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 0.6666666667z + 15 + z2 = 0 + 15

Reorder the terms:
-15 + 15 + 0.6666666667z + z2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 0.6666666667z + z2 = 0 + 15
0.6666666667z + z2 = 0 + 15

Combine like terms: 0 + 15 = 15
0.6666666667z + z2 = 15

The z term is 0.6666666667z.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667z + 0.1111111112 + z2 = 15 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667z + z2 = 15 + 0.1111111112

Combine like terms: 15 + 0.1111111112 = 15.1111111112
0.1111111112 + 0.6666666667z + z2 = 15.1111111112

Factor a perfect square on the left side:
(z + 0.3333333334)(z + 0.3333333334) = 15.1111111112

Calculate the square root of the right side: 3.887301263

Break this problem into two subproblems by setting 
(z + 0.3333333334) equal to 3.887301263 and -3.887301263.

Subproblem 1

z + 0.3333333334 = 3.887301263 Simplifying z + 0.3333333334 = 3.887301263 Reorder the terms: 0.3333333334 + z = 3.887301263 Solving 0.3333333334 + z = 3.887301263 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + z = 3.887301263 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + z = 3.887301263 + -0.3333333334 z = 3.887301263 + -0.3333333334 Combine like terms: 3.887301263 + -0.3333333334 = 3.5539679296 z = 3.5539679296 Simplifying z = 3.5539679296

Subproblem 2

z + 0.3333333334 = -3.887301263 Simplifying z + 0.3333333334 = -3.887301263 Reorder the terms: 0.3333333334 + z = -3.887301263 Solving 0.3333333334 + z = -3.887301263 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + z = -3.887301263 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + z = -3.887301263 + -0.3333333334 z = -3.887301263 + -0.3333333334 Combine like terms: -3.887301263 + -0.3333333334 = -4.2206345964 z = -4.2206345964 Simplifying z = -4.2206345964

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.5539679296, -4.2206345964}

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