x^2+2.5x+1=0

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Solution for x^2+2.5x+1=0 equation:


Simplifying
x2 + 2.5x + 1 = 0

Reorder the terms:
1 + 2.5x + x2 = 0

Solving
1 + 2.5x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 2.5x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + 2.5x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2.5x + x2 = 0 + -1
2.5x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
2.5x + x2 = -1

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 = -1 + 1.5625

Reorder the terms:
1.5625 + 2.5x + x2 = -1 + 1.5625

Combine like terms: -1 + 1.5625 = 0.5625
1.5625 + 2.5x + x2 = 0.5625

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 0.5625

Calculate the square root of the right side: 0.75

Break this problem into two subproblems by setting 
(x + 1.25) equal to 0.75 and -0.75.

Subproblem 1

x + 1.25 = 0.75 Simplifying x + 1.25 = 0.75 Reorder the terms: 1.25 + x = 0.75 Solving 1.25 + x = 0.75 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 = 0.75 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 0.75 + -1.25 x = 0.75 + -1.25 Combine like terms: 0.75 + -1.25 = -0.5 x = -0.5 Simplifying x = -0.5

Subproblem 2

x + 1.25 = -0.75 Simplifying x + 1.25 = -0.75 Reorder the terms: 1.25 + x = -0.75 Solving 1.25 + x = -0.75 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 = -0.75 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -0.75 + -1.25 x = -0.75 + -1.25 Combine like terms: -0.75 + -1.25 = -2 x = -2 Simplifying x = -2

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.5, -2}

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