4w^2+9w=5

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Solution for 4w^2+9w=5 equation:


Simplifying
4w2 + 9w = 5

Reorder the terms:
9w + 4w2 = 5

Solving
9w + 4w2 = 5

Solving for variable 'w'.

Reorder the terms:
-5 + 9w + 4w2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + 9w + 4w2 = 0

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

Divide each side by '4'.
-1.25 + 2.25w + w2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + 2.25w + 1.25 + w2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + 2.25w + w2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + 2.25w + w2 = 0 + 1.25
2.25w + w2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
2.25w + w2 = 1.25

The w term is 2.25w.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25w + 1.265625 + w2 = 1.25 + 1.265625

Reorder the terms:
1.265625 + 2.25w + w2 = 1.25 + 1.265625

Combine like terms: 1.25 + 1.265625 = 2.515625
1.265625 + 2.25w + w2 = 2.515625

Factor a perfect square on the left side:
(w + 1.125)(w + 1.125) = 2.515625

Calculate the square root of the right side: 1.586072193

Break this problem into two subproblems by setting 
(w + 1.125) equal to 1.586072193 and -1.586072193.

Subproblem 1

w + 1.125 = 1.586072193 Simplifying w + 1.125 = 1.586072193 Reorder the terms: 1.125 + w = 1.586072193 Solving 1.125 + w = 1.586072193 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + w = 1.586072193 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + w = 1.586072193 + -1.125 w = 1.586072193 + -1.125 Combine like terms: 1.586072193 + -1.125 = 0.461072193 w = 0.461072193 Simplifying w = 0.461072193

Subproblem 2

w + 1.125 = -1.586072193 Simplifying w + 1.125 = -1.586072193 Reorder the terms: 1.125 + w = -1.586072193 Solving 1.125 + w = -1.586072193 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + w = -1.586072193 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + w = -1.586072193 + -1.125 w = -1.586072193 + -1.125 Combine like terms: -1.586072193 + -1.125 = -2.711072193 w = -2.711072193 Simplifying w = -2.711072193

Solution

The solution to the problem is based on the solutions from the subproblems. w = {0.461072193, -2.711072193}

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