4x^2+10x+1=0

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Solution for 4x^2+10x+1=0 equation:


Simplifying
4x2 + 10x + 1 = 0

Reorder the terms:
1 + 10x + 4x2 = 0

Solving
1 + 10x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
0.25 + 2.5x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 2.5x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 2.5x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 2.5x + x2 = 0 + -0.25
2.5x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
2.5x + x2 = -0.25

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 = -0.25 + 1.5625

Reorder the terms:
1.5625 + 2.5x + x2 = -0.25 + 1.5625

Combine like terms: -0.25 + 1.5625 = 1.3125
1.5625 + 2.5x + x2 = 1.3125

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 1.3125

Calculate the square root of the right side: 1.145643924

Break this problem into two subproblems by setting 
(x + 1.25) equal to 1.145643924 and -1.145643924.

Subproblem 1

x + 1.25 = 1.145643924 Simplifying x + 1.25 = 1.145643924 Reorder the terms: 1.25 + x = 1.145643924 Solving 1.25 + x = 1.145643924 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 = 1.145643924 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 1.145643924 + -1.25 x = 1.145643924 + -1.25 Combine like terms: 1.145643924 + -1.25 = -0.104356076 x = -0.104356076 Simplifying x = -0.104356076

Subproblem 2

x + 1.25 = -1.145643924 Simplifying x + 1.25 = -1.145643924 Reorder the terms: 1.25 + x = -1.145643924 Solving 1.25 + x = -1.145643924 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 = -1.145643924 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -1.145643924 + -1.25 x = -1.145643924 + -1.25 Combine like terms: -1.145643924 + -1.25 = -2.395643924 x = -2.395643924 Simplifying x = -2.395643924

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.104356076, -2.395643924}

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