5w^2=-3w+1

Solution for 5w^2=-3w+1 equation:

Simplifying
5w2 = -3w + 1

Reorder the terms:
5w2 = 1 + -3w

Solving
5w2 = 1 + -3w

Solving for variable 'w'.

Reorder the terms:
-1 + 3w + 5w2 = 1 + -3w + -1 + 3w

Reorder the terms:
-1 + 3w + 5w2 = 1 + -1 + -3w + 3w

Combine like terms: 1 + -1 = 0
-1 + 3w + 5w2 = 0 + -3w + 3w
-1 + 3w + 5w2 = -3w + 3w

Combine like terms: -3w + 3w = 0
-1 + 3w + 5w2 = 0

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

Divide each side by '5'.
-0.2 + 0.6w + w2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + 0.6w + 0.2 + w2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + 0.6w + w2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + 0.6w + w2 = 0 + 0.2
0.6w + w2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
0.6w + w2 = 0.2

The w term is 0.6w.  Take half its coefficient (0.3).
Square it (0.09) and add it to both sides.

Add '0.09' to each side of the equation.
0.6w + 0.09 + w2 = 0.2 + 0.09

Reorder the terms:
0.09 + 0.6w + w2 = 0.2 + 0.09

Combine like terms: 0.2 + 0.09 = 0.29
0.09 + 0.6w + w2 = 0.29

Factor a perfect square on the left side:
(w + 0.3)(w + 0.3) = 0.29

Calculate the square root of the right side: 0.538516481

Break this problem into two subproblems by setting 
(w + 0.3) equal to 0.538516481 and -0.538516481.

Subproblem 1
w + 0.3 = 0.538516481

Simplifying
w + 0.3 = 0.538516481

Reorder the terms:
0.3 + w = 0.538516481

Solving
0.3 + w = 0.538516481

Solving for variable 'w'.

Move all terms containing w to the left, all other terms to the right.

Add '-0.3' to each side of the equation.
0.3 + -0.3 + w = 0.538516481 + -0.3

Combine like terms: 0.3 + -0.3 = 0.0
0.0 + w = 0.538516481 + -0.3
w = 0.538516481 + -0.3

Combine like terms: 0.538516481 + -0.3 = 0.238516481
w = 0.238516481

Simplifying
w = 0.238516481

Subproblem 2
w + 0.3 = -0.538516481

Simplifying
w + 0.3 = -0.538516481

Reorder the terms:
0.3 + w = -0.538516481

Solving
0.3 + w = -0.538516481

Solving for variable 'w'.

Move all terms containing w to the left, all other terms to the right.

Add '-0.3' to each side of the equation.
0.3 + -0.3 + w = -0.538516481 + -0.3

Combine like terms: 0.3 + -0.3 = 0.0
0.0 + w = -0.538516481 + -0.3
w = -0.538516481 + -0.3

Combine like terms: -0.538516481 + -0.3 = -0.838516481
w = -0.838516481

Simplifying
w = -0.838516481

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {0.238516481, -0.838516481}