0=3w^2+20w-45

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Solution for 0=3w^2+20w-45 equation:


Simplifying
0 = 3w2 + 20w + -45

Reorder the terms:
0 = -45 + 20w + 3w2

Solving
0 = -45 + 20w + 3w2

Solving for variable 'w'.

Combine like terms: 0 + 45 = 45
45 + -20w + -3w2 = -45 + 20w + 3w2 + 45 + -20w + -3w2

Reorder the terms:
45 + -20w + -3w2 = -45 + 45 + 20w + -20w + 3w2 + -3w2

Combine like terms: -45 + 45 = 0
45 + -20w + -3w2 = 0 + 20w + -20w + 3w2 + -3w2
45 + -20w + -3w2 = 20w + -20w + 3w2 + -3w2

Combine like terms: 20w + -20w = 0
45 + -20w + -3w2 = 0 + 3w2 + -3w2
45 + -20w + -3w2 = 3w2 + -3w2

Combine like terms: 3w2 + -3w2 = 0
45 + -20w + -3w2 = 0

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

Divide each side by '-3'.
-15 + 6.666666667w + w2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 6.666666667w + 15 + w2 = 0 + 15

Reorder the terms:
-15 + 15 + 6.666666667w + w2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 6.666666667w + w2 = 0 + 15
6.666666667w + w2 = 0 + 15

Combine like terms: 0 + 15 = 15
6.666666667w + w2 = 15

The w term is 6.666666667w.  Take half its coefficient (3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
6.666666667w + 11.11111112 + w2 = 15 + 11.11111112

Reorder the terms:
11.11111112 + 6.666666667w + w2 = 15 + 11.11111112

Combine like terms: 15 + 11.11111112 = 26.11111112
11.11111112 + 6.666666667w + w2 = 26.11111112

Factor a perfect square on the left side:
(w + 3.333333334)(w + 3.333333334) = 26.11111112

Calculate the square root of the right side: 5.10990324

Break this problem into two subproblems by setting 
(w + 3.333333334) equal to 5.10990324 and -5.10990324.

Subproblem 1

w + 3.333333334 = 5.10990324 Simplifying w + 3.333333334 = 5.10990324 Reorder the terms: 3.333333334 + w = 5.10990324 Solving 3.333333334 + w = 5.10990324 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-3.333333334' to each side of the equation. 3.333333334 + -3.333333334 + w = 5.10990324 + -3.333333334 Combine like terms: 3.333333334 + -3.333333334 = 0.000000000 0.000000000 + w = 5.10990324 + -3.333333334 w = 5.10990324 + -3.333333334 Combine like terms: 5.10990324 + -3.333333334 = 1.776569906 w = 1.776569906 Simplifying w = 1.776569906

Subproblem 2

w + 3.333333334 = -5.10990324 Simplifying w + 3.333333334 = -5.10990324 Reorder the terms: 3.333333334 + w = -5.10990324 Solving 3.333333334 + w = -5.10990324 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-3.333333334' to each side of the equation. 3.333333334 + -3.333333334 + w = -5.10990324 + -3.333333334 Combine like terms: 3.333333334 + -3.333333334 = 0.000000000 0.000000000 + w = -5.10990324 + -3.333333334 w = -5.10990324 + -3.333333334 Combine like terms: -5.10990324 + -3.333333334 = -8.443236574 w = -8.443236574 Simplifying w = -8.443236574

Solution

The solution to the problem is based on the solutions from the subproblems. w = {1.776569906, -8.443236574}

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