(4w^2-6w+3)+(-5w^2+w+7)=0

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Solution for (4w^2-6w+3)+(-5w^2+w+7)=0 equation:


Simplifying
(4w2 + -6w + 3) + (-5w2 + w + 7) = 0

Reorder the terms:
(3 + -6w + 4w2) + (-5w2 + w + 7) = 0

Remove parenthesis around (3 + -6w + 4w2)
3 + -6w + 4w2 + (-5w2 + w + 7) = 0

Reorder the terms:
3 + -6w + 4w2 + (7 + w + -5w2) = 0

Remove parenthesis around (7 + w + -5w2)
3 + -6w + 4w2 + 7 + w + -5w2 = 0

Reorder the terms:
3 + 7 + -6w + w + 4w2 + -5w2 = 0

Combine like terms: 3 + 7 = 10
10 + -6w + w + 4w2 + -5w2 = 0

Combine like terms: -6w + w = -5w
10 + -5w + 4w2 + -5w2 = 0

Combine like terms: 4w2 + -5w2 = -1w2
10 + -5w + -1w2 = 0

Solving
10 + -5w + -1w2 = 0

Solving for variable 'w'.

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

Divide each side by '-1'.
-10 + 5w + w2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 5w + 10 + w2 = 0 + 10

Reorder the terms:
-10 + 10 + 5w + w2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 5w + w2 = 0 + 10
5w + w2 = 0 + 10

Combine like terms: 0 + 10 = 10
5w + w2 = 10

The w term is 5w.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5w + 6.25 + w2 = 10 + 6.25

Reorder the terms:
6.25 + 5w + w2 = 10 + 6.25

Combine like terms: 10 + 6.25 = 16.25
6.25 + 5w + w2 = 16.25

Factor a perfect square on the left side:
(w + 2.5)(w + 2.5) = 16.25

Calculate the square root of the right side: 4.031128874

Break this problem into two subproblems by setting 
(w + 2.5) equal to 4.031128874 and -4.031128874.

Subproblem 1

w + 2.5 = 4.031128874 Simplifying w + 2.5 = 4.031128874 Reorder the terms: 2.5 + w = 4.031128874 Solving 2.5 + w = 4.031128874 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + w = 4.031128874 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + w = 4.031128874 + -2.5 w = 4.031128874 + -2.5 Combine like terms: 4.031128874 + -2.5 = 1.531128874 w = 1.531128874 Simplifying w = 1.531128874

Subproblem 2

w + 2.5 = -4.031128874 Simplifying w + 2.5 = -4.031128874 Reorder the terms: 2.5 + w = -4.031128874 Solving 2.5 + w = -4.031128874 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + w = -4.031128874 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + w = -4.031128874 + -2.5 w = -4.031128874 + -2.5 Combine like terms: -4.031128874 + -2.5 = -6.531128874 w = -6.531128874 Simplifying w = -6.531128874

Solution

The solution to the problem is based on the solutions from the subproblems. w = {1.531128874, -6.531128874}

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