4w^2+7w=194

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Solution for 4w^2+7w=194 equation:


Simplifying
4w2 + 7w = 194

Reorder the terms:
7w + 4w2 = 194

Solving
7w + 4w2 = 194

Solving for variable 'w'.

Reorder the terms:
-194 + 7w + 4w2 = 194 + -194

Combine like terms: 194 + -194 = 0
-194 + 7w + 4w2 = 0

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

Divide each side by '4'.
-48.5 + 1.75w + w2 = 0

Move the constant term to the right:

Add '48.5' to each side of the equation.
-48.5 + 1.75w + 48.5 + w2 = 0 + 48.5

Reorder the terms:
-48.5 + 48.5 + 1.75w + w2 = 0 + 48.5

Combine like terms: -48.5 + 48.5 = 0.0
0.0 + 1.75w + w2 = 0 + 48.5
1.75w + w2 = 0 + 48.5

Combine like terms: 0 + 48.5 = 48.5
1.75w + w2 = 48.5

The w term is 1.75w.  Take half its coefficient (0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
1.75w + 0.765625 + w2 = 48.5 + 0.765625

Reorder the terms:
0.765625 + 1.75w + w2 = 48.5 + 0.765625

Combine like terms: 48.5 + 0.765625 = 49.265625
0.765625 + 1.75w + w2 = 49.265625

Factor a perfect square on the left side:
(w + 0.875)(w + 0.875) = 49.265625

Calculate the square root of the right side: 7.018947571

Break this problem into two subproblems by setting 
(w + 0.875) equal to 7.018947571 and -7.018947571.

Subproblem 1

w + 0.875 = 7.018947571 Simplifying w + 0.875 = 7.018947571 Reorder the terms: 0.875 + w = 7.018947571 Solving 0.875 + w = 7.018947571 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + w = 7.018947571 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + w = 7.018947571 + -0.875 w = 7.018947571 + -0.875 Combine like terms: 7.018947571 + -0.875 = 6.143947571 w = 6.143947571 Simplifying w = 6.143947571

Subproblem 2

w + 0.875 = -7.018947571 Simplifying w + 0.875 = -7.018947571 Reorder the terms: 0.875 + w = -7.018947571 Solving 0.875 + w = -7.018947571 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + w = -7.018947571 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + w = -7.018947571 + -0.875 w = -7.018947571 + -0.875 Combine like terms: -7.018947571 + -0.875 = -7.893947571 w = -7.893947571 Simplifying w = -7.893947571

Solution

The solution to the problem is based on the solutions from the subproblems. w = {6.143947571, -7.893947571}

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