4w^2=-9w-1

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Solution for 4w^2=-9w-1 equation:


Simplifying
4w2 = -9w + -1

Reorder the terms:
4w2 = -1 + -9w

Solving
4w2 = -1 + -9w

Solving for variable 'w'.

Reorder the terms:
1 + 9w + 4w2 = -1 + -9w + 1 + 9w

Reorder the terms:
1 + 9w + 4w2 = -1 + 1 + -9w + 9w

Combine like terms: -1 + 1 = 0
1 + 9w + 4w2 = 0 + -9w + 9w
1 + 9w + 4w2 = -9w + 9w

Combine like terms: -9w + 9w = 0
1 + 9w + 4w2 = 0

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

Divide each side by '4'.
0.25 + 2.25w + w2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 2.25w + -0.25 + w2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 2.25w + w2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 2.25w + w2 = 0 + -0.25
2.25w + w2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
2.25w + w2 = -0.25

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 = -0.25 + 1.265625

Reorder the terms:
1.265625 + 2.25w + w2 = -0.25 + 1.265625

Combine like terms: -0.25 + 1.265625 = 1.015625
1.265625 + 2.25w + w2 = 1.015625

Factor a perfect square on the left side:
(w + 1.125)(w + 1.125) = 1.015625

Calculate the square root of the right side: 1.007782219

Break this problem into two subproblems by setting 
(w + 1.125) equal to 1.007782219 and -1.007782219.

Subproblem 1

w + 1.125 = 1.007782219 Simplifying w + 1.125 = 1.007782219 Reorder the terms: 1.125 + w = 1.007782219 Solving 1.125 + w = 1.007782219 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 = 1.007782219 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + w = 1.007782219 + -1.125 w = 1.007782219 + -1.125 Combine like terms: 1.007782219 + -1.125 = -0.117217781 w = -0.117217781 Simplifying w = -0.117217781

Subproblem 2

w + 1.125 = -1.007782219 Simplifying w + 1.125 = -1.007782219 Reorder the terms: 1.125 + w = -1.007782219 Solving 1.125 + w = -1.007782219 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 = -1.007782219 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + w = -1.007782219 + -1.125 w = -1.007782219 + -1.125 Combine like terms: -1.007782219 + -1.125 = -2.132782219 w = -2.132782219 Simplifying w = -2.132782219

Solution

The solution to the problem is based on the solutions from the subproblems. w = {-0.117217781, -2.132782219}

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