24x^2+12x+1=0

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Solution for 24x^2+12x+1=0 equation:


Simplifying
24x2 + 12x + 1 = 0

Reorder the terms:
1 + 12x + 24x2 = 0

Solving
1 + 12x + 24x2 = 0

Solving for variable 'x'.

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

Divide each side by '24'.
0.04166666667 + 0.5x + x2 = 0

Move the constant term to the right:

Add '-0.04166666667' to each side of the equation.
0.04166666667 + 0.5x + -0.04166666667 + x2 = 0 + -0.04166666667

Reorder the terms:
0.04166666667 + -0.04166666667 + 0.5x + x2 = 0 + -0.04166666667

Combine like terms: 0.04166666667 + -0.04166666667 = 0.00000000000
0.00000000000 + 0.5x + x2 = 0 + -0.04166666667
0.5x + x2 = 0 + -0.04166666667

Combine like terms: 0 + -0.04166666667 = -0.04166666667
0.5x + x2 = -0.04166666667

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = -0.04166666667 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = -0.04166666667 + 0.0625

Combine like terms: -0.04166666667 + 0.0625 = 0.02083333333
0.0625 + 0.5x + x2 = 0.02083333333

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 0.02083333333

Calculate the square root of the right side: 0.144337567

Break this problem into two subproblems by setting 
(x + 0.25) equal to 0.144337567 and -0.144337567.

Subproblem 1

x + 0.25 = 0.144337567 Simplifying x + 0.25 = 0.144337567 Reorder the terms: 0.25 + x = 0.144337567 Solving 0.25 + x = 0.144337567 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 0.144337567 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 0.144337567 + -0.25 x = 0.144337567 + -0.25 Combine like terms: 0.144337567 + -0.25 = -0.105662433 x = -0.105662433 Simplifying x = -0.105662433

Subproblem 2

x + 0.25 = -0.144337567 Simplifying x + 0.25 = -0.144337567 Reorder the terms: 0.25 + x = -0.144337567 Solving 0.25 + x = -0.144337567 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -0.144337567 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -0.144337567 + -0.25 x = -0.144337567 + -0.25 Combine like terms: -0.144337567 + -0.25 = -0.394337567 x = -0.394337567 Simplifying x = -0.394337567

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.105662433, -0.394337567}

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