w^2+3w=77

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


Simplifying
w2 + 3w = 77

Reorder the terms:
3w + w2 = 77

Solving
3w + w2 = 77

Solving for variable 'w'.

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

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

Begin completing the square.

Move the constant term to the right:

Add '77' to each side of the equation.
-77 + 3w + 77 + w2 = 0 + 77

Reorder the terms:
-77 + 77 + 3w + w2 = 0 + 77

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

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

The w term is 3w.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3w + 2.25 + w2 = 77 + 2.25

Reorder the terms:
2.25 + 3w + w2 = 77 + 2.25

Combine like terms: 77 + 2.25 = 79.25
2.25 + 3w + w2 = 79.25

Factor a perfect square on the left side:
(w + 1.5)(w + 1.5) = 79.25

Calculate the square root of the right side: 8.902246907

Break this problem into two subproblems by setting 
(w + 1.5) equal to 8.902246907 and -8.902246907.

Subproblem 1

w + 1.5 = 8.902246907 Simplifying w + 1.5 = 8.902246907 Reorder the terms: 1.5 + w = 8.902246907 Solving 1.5 + w = 8.902246907 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = 8.902246907 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 8.902246907 + -1.5 w = 8.902246907 + -1.5 Combine like terms: 8.902246907 + -1.5 = 7.402246907 w = 7.402246907 Simplifying w = 7.402246907

Subproblem 2

w + 1.5 = -8.902246907 Simplifying w + 1.5 = -8.902246907 Reorder the terms: 1.5 + w = -8.902246907 Solving 1.5 + w = -8.902246907 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = -8.902246907 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -8.902246907 + -1.5 w = -8.902246907 + -1.5 Combine like terms: -8.902246907 + -1.5 = -10.402246907 w = -10.402246907 Simplifying w = -10.402246907

Solution

The solution to the problem is based on the solutions from the subproblems. w = {7.402246907, -10.402246907}

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