3w^2+11w=77

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Solution for 3w^2+11w=77 equation:


Simplifying
3w2 + 11w = 77

Reorder the terms:
11w + 3w2 = 77

Solving
11w + 3w2 = 77

Solving for variable 'w'.

Reorder the terms:
-77 + 11w + 3w2 = 77 + -77

Combine like terms: 77 + -77 = 0
-77 + 11w + 3w2 = 0

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

Divide each side by '3'.
-25.66666667 + 3.666666667w + w2 = 0

Move the constant term to the right:

Add '25.66666667' to each side of the equation.
-25.66666667 + 3.666666667w + 25.66666667 + w2 = 0 + 25.66666667

Reorder the terms:
-25.66666667 + 25.66666667 + 3.666666667w + w2 = 0 + 25.66666667

Combine like terms: -25.66666667 + 25.66666667 = 0.00000000
0.00000000 + 3.666666667w + w2 = 0 + 25.66666667
3.666666667w + w2 = 0 + 25.66666667

Combine like terms: 0 + 25.66666667 = 25.66666667
3.666666667w + w2 = 25.66666667

The w term is 3.666666667w.  Take half its coefficient (1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
3.666666667w + 3.361111114 + w2 = 25.66666667 + 3.361111114

Reorder the terms:
3.361111114 + 3.666666667w + w2 = 25.66666667 + 3.361111114

Combine like terms: 25.66666667 + 3.361111114 = 29.027777784
3.361111114 + 3.666666667w + w2 = 29.027777784

Factor a perfect square on the left side:
(w + 1.833333334)(w + 1.833333334) = 29.027777784

Calculate the square root of the right side: 5.387743292

Break this problem into two subproblems by setting 
(w + 1.833333334) equal to 5.387743292 and -5.387743292.

Subproblem 1

w + 1.833333334 = 5.387743292 Simplifying w + 1.833333334 = 5.387743292 Reorder the terms: 1.833333334 + w = 5.387743292 Solving 1.833333334 + w = 5.387743292 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + w = 5.387743292 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + w = 5.387743292 + -1.833333334 w = 5.387743292 + -1.833333334 Combine like terms: 5.387743292 + -1.833333334 = 3.554409958 w = 3.554409958 Simplifying w = 3.554409958

Subproblem 2

w + 1.833333334 = -5.387743292 Simplifying w + 1.833333334 = -5.387743292 Reorder the terms: 1.833333334 + w = -5.387743292 Solving 1.833333334 + w = -5.387743292 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + w = -5.387743292 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + w = -5.387743292 + -1.833333334 w = -5.387743292 + -1.833333334 Combine like terms: -5.387743292 + -1.833333334 = -7.221076626 w = -7.221076626 Simplifying w = -7.221076626

Solution

The solution to the problem is based on the solutions from the subproblems. w = {3.554409958, -7.221076626}

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