2w^2-10w+1=w^2+16

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Solution for 2w^2-10w+1=w^2+16 equation:


Simplifying
2w2 + -10w + 1 = w2 + 16

Reorder the terms:
1 + -10w + 2w2 = w2 + 16

Reorder the terms:
1 + -10w + 2w2 = 16 + w2

Solving
1 + -10w + 2w2 = 16 + w2

Solving for variable 'w'.

Reorder the terms:
1 + -16 + -10w + 2w2 + -1w2 = 16 + w2 + -16 + -1w2

Combine like terms: 1 + -16 = -15
-15 + -10w + 2w2 + -1w2 = 16 + w2 + -16 + -1w2

Combine like terms: 2w2 + -1w2 = 1w2
-15 + -10w + 1w2 = 16 + w2 + -16 + -1w2

Reorder the terms:
-15 + -10w + 1w2 = 16 + -16 + w2 + -1w2

Combine like terms: 16 + -16 = 0
-15 + -10w + 1w2 = 0 + w2 + -1w2
-15 + -10w + 1w2 = w2 + -1w2

Combine like terms: w2 + -1w2 = 0
-15 + -10w + 1w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + -10w + 15 + w2 = 0 + 15

Reorder the terms:
-15 + 15 + -10w + w2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + -10w + w2 = 0 + 15
-10w + w2 = 0 + 15

Combine like terms: 0 + 15 = 15
-10w + w2 = 15

The w term is -10w.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10w + 25 + w2 = 15 + 25

Reorder the terms:
25 + -10w + w2 = 15 + 25

Combine like terms: 15 + 25 = 40
25 + -10w + w2 = 40

Factor a perfect square on the left side:
(w + -5)(w + -5) = 40

Calculate the square root of the right side: 6.32455532

Break this problem into two subproblems by setting 
(w + -5) equal to 6.32455532 and -6.32455532.

Subproblem 1

w + -5 = 6.32455532 Simplifying w + -5 = 6.32455532 Reorder the terms: -5 + w = 6.32455532 Solving -5 + w = 6.32455532 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + w = 6.32455532 + 5 Combine like terms: -5 + 5 = 0 0 + w = 6.32455532 + 5 w = 6.32455532 + 5 Combine like terms: 6.32455532 + 5 = 11.32455532 w = 11.32455532 Simplifying w = 11.32455532

Subproblem 2

w + -5 = -6.32455532 Simplifying w + -5 = -6.32455532 Reorder the terms: -5 + w = -6.32455532 Solving -5 + w = -6.32455532 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + w = -6.32455532 + 5 Combine like terms: -5 + 5 = 0 0 + w = -6.32455532 + 5 w = -6.32455532 + 5 Combine like terms: -6.32455532 + 5 = -1.32455532 w = -1.32455532 Simplifying w = -1.32455532

Solution

The solution to the problem is based on the solutions from the subproblems. w = {11.32455532, -1.32455532}

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