2w^2-19w+31=49-4w^2-4

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Solution for 2w^2-19w+31=49-4w^2-4 equation:


Simplifying
2w2 + -19w + 31 = 49 + -4w2 + -4

Reorder the terms:
31 + -19w + 2w2 = 49 + -4w2 + -4

Reorder the terms:
31 + -19w + 2w2 = 49 + -4 + -4w2

Combine like terms: 49 + -4 = 45
31 + -19w + 2w2 = 45 + -4w2

Solving
31 + -19w + 2w2 = 45 + -4w2

Solving for variable 'w'.

Reorder the terms:
31 + -45 + -19w + 2w2 + 4w2 = 45 + -4w2 + -45 + 4w2

Combine like terms: 31 + -45 = -14
-14 + -19w + 2w2 + 4w2 = 45 + -4w2 + -45 + 4w2

Combine like terms: 2w2 + 4w2 = 6w2
-14 + -19w + 6w2 = 45 + -4w2 + -45 + 4w2

Reorder the terms:
-14 + -19w + 6w2 = 45 + -45 + -4w2 + 4w2

Combine like terms: 45 + -45 = 0
-14 + -19w + 6w2 = 0 + -4w2 + 4w2
-14 + -19w + 6w2 = -4w2 + 4w2

Combine like terms: -4w2 + 4w2 = 0
-14 + -19w + 6w2 = 0

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

Divide each side by '6'.
-2.333333333 + -3.166666667w + w2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -3.166666667w + 2.333333333 + w2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -3.166666667w + w2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -3.166666667w + w2 = 0 + 2.333333333
-3.166666667w + w2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-3.166666667w + w2 = 2.333333333

The w term is -3.166666667w.  Take half its coefficient (-1.583333334).
Square it (2.506944447) and add it to both sides.

Add '2.506944447' to each side of the equation.
-3.166666667w + 2.506944447 + w2 = 2.333333333 + 2.506944447

Reorder the terms:
2.506944447 + -3.166666667w + w2 = 2.333333333 + 2.506944447

Combine like terms: 2.333333333 + 2.506944447 = 4.84027778
2.506944447 + -3.166666667w + w2 = 4.84027778

Factor a perfect square on the left side:
(w + -1.583333334)(w + -1.583333334) = 4.84027778

Calculate the square root of the right side: 2.200063131

Break this problem into two subproblems by setting 
(w + -1.583333334) equal to 2.200063131 and -2.200063131.

Subproblem 1

w + -1.583333334 = 2.200063131 Simplifying w + -1.583333334 = 2.200063131 Reorder the terms: -1.583333334 + w = 2.200063131 Solving -1.583333334 + w = 2.200063131 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.583333334' to each side of the equation. -1.583333334 + 1.583333334 + w = 2.200063131 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + w = 2.200063131 + 1.583333334 w = 2.200063131 + 1.583333334 Combine like terms: 2.200063131 + 1.583333334 = 3.783396465 w = 3.783396465 Simplifying w = 3.783396465

Subproblem 2

w + -1.583333334 = -2.200063131 Simplifying w + -1.583333334 = -2.200063131 Reorder the terms: -1.583333334 + w = -2.200063131 Solving -1.583333334 + w = -2.200063131 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.583333334' to each side of the equation. -1.583333334 + 1.583333334 + w = -2.200063131 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + w = -2.200063131 + 1.583333334 w = -2.200063131 + 1.583333334 Combine like terms: -2.200063131 + 1.583333334 = -0.616729797 w = -0.616729797 Simplifying w = -0.616729797

Solution

The solution to the problem is based on the solutions from the subproblems. w = {3.783396465, -0.616729797}

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