3w-(w-5)(w-5)=2w+1

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Solution for 3w-(w-5)(w-5)=2w+1 equation:


Simplifying
3w + -1(w + -5)(w + -5) = 2w + 1

Reorder the terms:
3w + -1(-5 + w)(w + -5) = 2w + 1

Reorder the terms:
3w + -1(-5 + w)(-5 + w) = 2w + 1

Multiply (-5 + w) * (-5 + w)
3w + -1(-5(-5 + w) + w(-5 + w)) = 2w + 1
3w + -1((-5 * -5 + w * -5) + w(-5 + w)) = 2w + 1
3w + -1((25 + -5w) + w(-5 + w)) = 2w + 1
3w + -1(25 + -5w + (-5 * w + w * w)) = 2w + 1
3w + -1(25 + -5w + (-5w + w2)) = 2w + 1

Combine like terms: -5w + -5w = -10w
3w + -1(25 + -10w + w2) = 2w + 1
3w + (25 * -1 + -10w * -1 + w2 * -1) = 2w + 1
3w + (-25 + 10w + -1w2) = 2w + 1

Reorder the terms:
-25 + 3w + 10w + -1w2 = 2w + 1

Combine like terms: 3w + 10w = 13w
-25 + 13w + -1w2 = 2w + 1

Reorder the terms:
-25 + 13w + -1w2 = 1 + 2w

Solving
-25 + 13w + -1w2 = 1 + 2w

Solving for variable 'w'.

Reorder the terms:
-25 + -1 + 13w + -2w + -1w2 = 1 + 2w + -1 + -2w

Combine like terms: -25 + -1 = -26
-26 + 13w + -2w + -1w2 = 1 + 2w + -1 + -2w

Combine like terms: 13w + -2w = 11w
-26 + 11w + -1w2 = 1 + 2w + -1 + -2w

Reorder the terms:
-26 + 11w + -1w2 = 1 + -1 + 2w + -2w

Combine like terms: 1 + -1 = 0
-26 + 11w + -1w2 = 0 + 2w + -2w
-26 + 11w + -1w2 = 2w + -2w

Combine like terms: 2w + -2w = 0
-26 + 11w + -1w2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
26 + -11w + w2 = 0

Move the constant term to the right:

Add '-26' to each side of the equation.
26 + -11w + -26 + w2 = 0 + -26

Reorder the terms:
26 + -26 + -11w + w2 = 0 + -26

Combine like terms: 26 + -26 = 0
0 + -11w + w2 = 0 + -26
-11w + w2 = 0 + -26

Combine like terms: 0 + -26 = -26
-11w + w2 = -26

The w term is -11w.  Take half its coefficient (-5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
-11w + 30.25 + w2 = -26 + 30.25

Reorder the terms:
30.25 + -11w + w2 = -26 + 30.25

Combine like terms: -26 + 30.25 = 4.25
30.25 + -11w + w2 = 4.25

Factor a perfect square on the left side:
(w + -5.5)(w + -5.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(w + -5.5) equal to 2.061552813 and -2.061552813.

Subproblem 1

w + -5.5 = 2.061552813 Simplifying w + -5.5 = 2.061552813 Reorder the terms: -5.5 + w = 2.061552813 Solving -5.5 + w = 2.061552813 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + w = 2.061552813 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + w = 2.061552813 + 5.5 w = 2.061552813 + 5.5 Combine like terms: 2.061552813 + 5.5 = 7.561552813 w = 7.561552813 Simplifying w = 7.561552813

Subproblem 2

w + -5.5 = -2.061552813 Simplifying w + -5.5 = -2.061552813 Reorder the terms: -5.5 + w = -2.061552813 Solving -5.5 + w = -2.061552813 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + w = -2.061552813 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + w = -2.061552813 + 5.5 w = -2.061552813 + 5.5 Combine like terms: -2.061552813 + 5.5 = 3.438447187 w = 3.438447187 Simplifying w = 3.438447187

Solution

The solution to the problem is based on the solutions from the subproblems. w = {7.561552813, 3.438447187}

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