0=7w^2+30w-22

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Solution for 0=7w^2+30w-22 equation:


Simplifying
0 = 7w2 + 30w + -22

Reorder the terms:
0 = -22 + 30w + 7w2

Solving
0 = -22 + 30w + 7w2

Solving for variable 'w'.

Combine like terms: 0 + 22 = 22
22 + -30w + -7w2 = -22 + 30w + 7w2 + 22 + -30w + -7w2

Reorder the terms:
22 + -30w + -7w2 = -22 + 22 + 30w + -30w + 7w2 + -7w2

Combine like terms: -22 + 22 = 0
22 + -30w + -7w2 = 0 + 30w + -30w + 7w2 + -7w2
22 + -30w + -7w2 = 30w + -30w + 7w2 + -7w2

Combine like terms: 30w + -30w = 0
22 + -30w + -7w2 = 0 + 7w2 + -7w2
22 + -30w + -7w2 = 7w2 + -7w2

Combine like terms: 7w2 + -7w2 = 0
22 + -30w + -7w2 = 0

Begin completing the square.  Divide all terms by
-7 the coefficient of the squared term: 

Divide each side by '-7'.
-3.142857143 + 4.285714286w + w2 = 0

Move the constant term to the right:

Add '3.142857143' to each side of the equation.
-3.142857143 + 4.285714286w + 3.142857143 + w2 = 0 + 3.142857143

Reorder the terms:
-3.142857143 + 3.142857143 + 4.285714286w + w2 = 0 + 3.142857143

Combine like terms: -3.142857143 + 3.142857143 = 0.000000000
0.000000000 + 4.285714286w + w2 = 0 + 3.142857143
4.285714286w + w2 = 0 + 3.142857143

Combine like terms: 0 + 3.142857143 = 3.142857143
4.285714286w + w2 = 3.142857143

The w term is 4.285714286w.  Take half its coefficient (2.142857143).
Square it (4.591836735) and add it to both sides.

Add '4.591836735' to each side of the equation.
4.285714286w + 4.591836735 + w2 = 3.142857143 + 4.591836735

Reorder the terms:
4.591836735 + 4.285714286w + w2 = 3.142857143 + 4.591836735

Combine like terms: 3.142857143 + 4.591836735 = 7.734693878
4.591836735 + 4.285714286w + w2 = 7.734693878

Factor a perfect square on the left side:
(w + 2.142857143)(w + 2.142857143) = 7.734693878

Calculate the square root of the right side: 2.781131762

Break this problem into two subproblems by setting 
(w + 2.142857143) equal to 2.781131762 and -2.781131762.

Subproblem 1

w + 2.142857143 = 2.781131762 Simplifying w + 2.142857143 = 2.781131762 Reorder the terms: 2.142857143 + w = 2.781131762 Solving 2.142857143 + w = 2.781131762 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.142857143' to each side of the equation. 2.142857143 + -2.142857143 + w = 2.781131762 + -2.142857143 Combine like terms: 2.142857143 + -2.142857143 = 0.000000000 0.000000000 + w = 2.781131762 + -2.142857143 w = 2.781131762 + -2.142857143 Combine like terms: 2.781131762 + -2.142857143 = 0.638274619 w = 0.638274619 Simplifying w = 0.638274619

Subproblem 2

w + 2.142857143 = -2.781131762 Simplifying w + 2.142857143 = -2.781131762 Reorder the terms: 2.142857143 + w = -2.781131762 Solving 2.142857143 + w = -2.781131762 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.142857143' to each side of the equation. 2.142857143 + -2.142857143 + w = -2.781131762 + -2.142857143 Combine like terms: 2.142857143 + -2.142857143 = 0.000000000 0.000000000 + w = -2.781131762 + -2.142857143 w = -2.781131762 + -2.142857143 Combine like terms: -2.781131762 + -2.142857143 = -4.923988905 w = -4.923988905 Simplifying w = -4.923988905

Solution

The solution to the problem is based on the solutions from the subproblems. w = {0.638274619, -4.923988905}

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