4x^2+14x+1=0

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Solution for 4x^2+14x+1=0 equation:


Simplifying
4x2 + 14x + 1 = 0

Reorder the terms:
1 + 14x + 4x2 = 0

Solving
1 + 14x + 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 + 3.5x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 3.5x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 3.5x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 3.5x + x2 = 0 + -0.25
3.5x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
3.5x + x2 = -0.25

The x term is 3.5x.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5x + 3.0625 + x2 = -0.25 + 3.0625

Reorder the terms:
3.0625 + 3.5x + x2 = -0.25 + 3.0625

Combine like terms: -0.25 + 3.0625 = 2.8125
3.0625 + 3.5x + x2 = 2.8125

Factor a perfect square on the left side:
(x + 1.75)(x + 1.75) = 2.8125

Calculate the square root of the right side: 1.677050983

Break this problem into two subproblems by setting 
(x + 1.75) equal to 1.677050983 and -1.677050983.

Subproblem 1

x + 1.75 = 1.677050983 Simplifying x + 1.75 = 1.677050983 Reorder the terms: 1.75 + x = 1.677050983 Solving 1.75 + x = 1.677050983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 1.677050983 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 1.677050983 + -1.75 x = 1.677050983 + -1.75 Combine like terms: 1.677050983 + -1.75 = -0.072949017 x = -0.072949017 Simplifying x = -0.072949017

Subproblem 2

x + 1.75 = -1.677050983 Simplifying x + 1.75 = -1.677050983 Reorder the terms: 1.75 + x = -1.677050983 Solving 1.75 + x = -1.677050983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -1.677050983 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -1.677050983 + -1.75 x = -1.677050983 + -1.75 Combine like terms: -1.677050983 + -1.75 = -3.427050983 x = -3.427050983 Simplifying x = -3.427050983

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.072949017, -3.427050983}

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