2x^2+5x+1=0

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Solution for 2x^2+5x+1=0 equation:


Simplifying
2x2 + 5x + 1 = 0

Reorder the terms:
1 + 5x + 2x2 = 0

Solving
1 + 5x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
0.5 + 2.5x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2.5x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2.5x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2.5x + x2 = 0 + -0.5
2.5x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2.5x + x2 = -0.5

The x term is 2.5x.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5x + 1.5625 + x2 = -0.5 + 1.5625

Reorder the terms:
1.5625 + 2.5x + x2 = -0.5 + 1.5625

Combine like terms: -0.5 + 1.5625 = 1.0625
1.5625 + 2.5x + x2 = 1.0625

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(x + 1.25) equal to 1.030776406 and -1.030776406.

Subproblem 1

x + 1.25 = 1.030776406 Simplifying x + 1.25 = 1.030776406 Reorder the terms: 1.25 + x = 1.030776406 Solving 1.25 + x = 1.030776406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = 1.030776406 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 1.030776406 + -1.25 x = 1.030776406 + -1.25 Combine like terms: 1.030776406 + -1.25 = -0.219223594 x = -0.219223594 Simplifying x = -0.219223594

Subproblem 2

x + 1.25 = -1.030776406 Simplifying x + 1.25 = -1.030776406 Reorder the terms: 1.25 + x = -1.030776406 Solving 1.25 + x = -1.030776406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = -1.030776406 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -1.030776406 + -1.25 x = -1.030776406 + -1.25 Combine like terms: -1.030776406 + -1.25 = -2.280776406 x = -2.280776406 Simplifying x = -2.280776406

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.219223594, -2.280776406}

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