4x^2+22x+5=0

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Solution for 4x^2+22x+5=0 equation:


Simplifying
4x2 + 22x + 5 = 0

Reorder the terms:
5 + 22x + 4x2 = 0

Solving
5 + 22x + 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'.
1.25 + 5.5x + x2 = 0

Move the constant term to the right:

Add '-1.25' to each side of the equation.
1.25 + 5.5x + -1.25 + x2 = 0 + -1.25

Reorder the terms:
1.25 + -1.25 + 5.5x + x2 = 0 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + 5.5x + x2 = 0 + -1.25
5.5x + x2 = 0 + -1.25

Combine like terms: 0 + -1.25 = -1.25
5.5x + x2 = -1.25

The x term is 5.5x.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5x + 7.5625 + x2 = -1.25 + 7.5625

Reorder the terms:
7.5625 + 5.5x + x2 = -1.25 + 7.5625

Combine like terms: -1.25 + 7.5625 = 6.3125
7.5625 + 5.5x + x2 = 6.3125

Factor a perfect square on the left side:
(x + 2.75)(x + 2.75) = 6.3125

Calculate the square root of the right side: 2.512468905

Break this problem into two subproblems by setting 
(x + 2.75) equal to 2.512468905 and -2.512468905.

Subproblem 1

x + 2.75 = 2.512468905 Simplifying x + 2.75 = 2.512468905 Reorder the terms: 2.75 + x = 2.512468905 Solving 2.75 + x = 2.512468905 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = 2.512468905 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = 2.512468905 + -2.75 x = 2.512468905 + -2.75 Combine like terms: 2.512468905 + -2.75 = -0.237531095 x = -0.237531095 Simplifying x = -0.237531095

Subproblem 2

x + 2.75 = -2.512468905 Simplifying x + 2.75 = -2.512468905 Reorder the terms: 2.75 + x = -2.512468905 Solving 2.75 + x = -2.512468905 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = -2.512468905 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = -2.512468905 + -2.75 x = -2.512468905 + -2.75 Combine like terms: -2.512468905 + -2.75 = -5.262468905 x = -5.262468905 Simplifying x = -5.262468905

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.237531095, -5.262468905}

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