7w^2+9w-7=0

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Solution for 7w^2+9w-7=0 equation:


Simplifying
7w2 + 9w + -7 = 0

Reorder the terms:
-7 + 9w + 7w2 = 0

Solving
-7 + 9w + 7w2 = 0

Solving for variable 'w'.

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

Divide each side by '7'.
-1 + 1.285714286w + w2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 1.285714286w + 1 + w2 = 0 + 1

Reorder the terms:
-1 + 1 + 1.285714286w + w2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 1.285714286w + w2 = 0 + 1
1.285714286w + w2 = 0 + 1

Combine like terms: 0 + 1 = 1
1.285714286w + w2 = 1

The w term is 1.285714286w.  Take half its coefficient (0.642857143).
Square it (0.4132653063) and add it to both sides.

Add '0.4132653063' to each side of the equation.
1.285714286w + 0.4132653063 + w2 = 1 + 0.4132653063

Reorder the terms:
0.4132653063 + 1.285714286w + w2 = 1 + 0.4132653063

Combine like terms: 1 + 0.4132653063 = 1.4132653063
0.4132653063 + 1.285714286w + w2 = 1.4132653063

Factor a perfect square on the left side:
(w + 0.642857143)(w + 0.642857143) = 1.4132653063

Calculate the square root of the right side: 1.188808356

Break this problem into two subproblems by setting 
(w + 0.642857143) equal to 1.188808356 and -1.188808356.

Subproblem 1

w + 0.642857143 = 1.188808356 Simplifying w + 0.642857143 = 1.188808356 Reorder the terms: 0.642857143 + w = 1.188808356 Solving 0.642857143 + w = 1.188808356 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.642857143' to each side of the equation. 0.642857143 + -0.642857143 + w = 1.188808356 + -0.642857143 Combine like terms: 0.642857143 + -0.642857143 = 0.000000000 0.000000000 + w = 1.188808356 + -0.642857143 w = 1.188808356 + -0.642857143 Combine like terms: 1.188808356 + -0.642857143 = 0.545951213 w = 0.545951213 Simplifying w = 0.545951213

Subproblem 2

w + 0.642857143 = -1.188808356 Simplifying w + 0.642857143 = -1.188808356 Reorder the terms: 0.642857143 + w = -1.188808356 Solving 0.642857143 + w = -1.188808356 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.642857143' to each side of the equation. 0.642857143 + -0.642857143 + w = -1.188808356 + -0.642857143 Combine like terms: 0.642857143 + -0.642857143 = 0.000000000 0.000000000 + w = -1.188808356 + -0.642857143 w = -1.188808356 + -0.642857143 Combine like terms: -1.188808356 + -0.642857143 = -1.831665499 w = -1.831665499 Simplifying w = -1.831665499

Solution

The solution to the problem is based on the solutions from the subproblems. w = {0.545951213, -1.831665499}

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