16x^2+24x+7=0

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Solution for 16x^2+24x+7=0 equation: 


Simplifying
16x2 + 24x + 7 = 0

Reorder the terms:
7 + 24x + 16x2 = 0

Solving
7 + 24x + 16x2 = 0

Solving for variable 'x'.

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

Divide each side by '16'.
0.4375 + 1.5x + x2 = 0

Move the constant term to the right:

Add '-0.4375' to each side of the equation.
0.4375 + 1.5x + -0.4375 + x2 = 0 + -0.4375

Reorder the terms:
0.4375 + -0.4375 + 1.5x + x2 = 0 + -0.4375

Combine like terms: 0.4375 + -0.4375 = 0.0000
0.0000 + 1.5x + x2 = 0 + -0.4375
1.5x + x2 = 0 + -0.4375

Combine like terms: 0 + -0.4375 = -0.4375
1.5x + x2 = -0.4375

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = -0.4375 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = -0.4375 + 0.5625

Combine like terms: -0.4375 + 0.5625 = 0.125
0.5625 + 1.5x + x2 = 0.125

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 0.125

Calculate the square root of the right side: 0.353553391

Break this problem into two subproblems by setting 
(x + 0.75) equal to 0.353553391 and -0.353553391.


Subproblem 1
x + 0.75 = 0.353553391

Simplifying
x + 0.75 = 0.353553391

Reorder the terms:
0.75 + x = 0.353553391

Solving
0.75 + x = 0.353553391

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = 0.353553391 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 0.353553391 + -0.75
x = 0.353553391 + -0.75

Combine like terms: 0.353553391 + -0.75 = -0.396446609
x = -0.396446609

Simplifying
x = -0.396446609


Subproblem 2
x + 0.75 = -0.353553391

Simplifying
x + 0.75 = -0.353553391

Reorder the terms:
0.75 + x = -0.353553391

Solving
0.75 + x = -0.353553391

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = -0.353553391 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -0.353553391 + -0.75
x = -0.353553391 + -0.75

Combine like terms: -0.353553391 + -0.75 = -1.103553391
x = -1.103553391

Simplifying
x = -1.103553391


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.396446609, -1.103553391}

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