4x^2+22x+15=0

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Solution for 4x^2+22x+15=0 equation: 


Simplifying
4x2 + 22x + 15 = 0

Reorder the terms:
15 + 22x + 4x2 = 0

Solving
15 + 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'.
3.75 + 5.5x + x2 = 0

Move the constant term to the right:

Add '-3.75' to each side of the equation.
3.75 + 5.5x + -3.75 + x2 = 0 + -3.75

Reorder the terms:
3.75 + -3.75 + 5.5x + x2 = 0 + -3.75

Combine like terms: 3.75 + -3.75 = 0.00
0.00 + 5.5x + x2 = 0 + -3.75
5.5x + x2 = 0 + -3.75

Combine like terms: 0 + -3.75 = -3.75
5.5x + x2 = -3.75

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 = -3.75 + 7.5625

Reorder the terms:
7.5625 + 5.5x + x2 = -3.75 + 7.5625

Combine like terms: -3.75 + 7.5625 = 3.8125
7.5625 + 5.5x + x2 = 3.8125

Factor a perfect square on the left side:
(x + 2.75)(x + 2.75) = 3.8125

Calculate the square root of the right side: 1.952562419

Break this problem into two subproblems by setting 
(x + 2.75) equal to 1.952562419 and -1.952562419.


Subproblem 1
x + 2.75 = 1.952562419

Simplifying
x + 2.75 = 1.952562419

Reorder the terms:
2.75 + x = 1.952562419

Solving
2.75 + x = 1.952562419

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 = 1.952562419 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = 1.952562419 + -2.75
x = 1.952562419 + -2.75

Combine like terms: 1.952562419 + -2.75 = -0.797437581
x = -0.797437581

Simplifying
x = -0.797437581


Subproblem 2
x + 2.75 = -1.952562419

Simplifying
x + 2.75 = -1.952562419

Reorder the terms:
2.75 + x = -1.952562419

Solving
2.75 + x = -1.952562419

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 = -1.952562419 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = -1.952562419 + -2.75
x = -1.952562419 + -2.75

Combine like terms: -1.952562419 + -2.75 = -4.702562419
x = -4.702562419

Simplifying
x = -4.702562419


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.797437581, -4.702562419}

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