24x^2-46x-1=0

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Solution for 24x^2-46x-1=0 equation:


Simplifying
24x2 + -46x + -1 = 0

Reorder the terms:
-1 + -46x + 24x2 = 0

Solving
-1 + -46x + 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 + -1.916666667x + x2 = 0

Move the constant term to the right:

Add '0.04166666667' to each side of the equation.
-0.04166666667 + -1.916666667x + 0.04166666667 + x2 = 0 + 0.04166666667

Reorder the terms:
-0.04166666667 + 0.04166666667 + -1.916666667x + x2 = 0 + 0.04166666667

Combine like terms: -0.04166666667 + 0.04166666667 = 0.00000000000
0.00000000000 + -1.916666667x + x2 = 0 + 0.04166666667
-1.916666667x + x2 = 0 + 0.04166666667

Combine like terms: 0 + 0.04166666667 = 0.04166666667
-1.916666667x + x2 = 0.04166666667

The x term is -1.916666667x.  Take half its coefficient (-0.9583333335).
Square it (0.9184027781) and add it to both sides.

Add '0.9184027781' to each side of the equation.
-1.916666667x + 0.9184027781 + x2 = 0.04166666667 + 0.9184027781

Reorder the terms:
0.9184027781 + -1.916666667x + x2 = 0.04166666667 + 0.9184027781

Combine like terms: 0.04166666667 + 0.9184027781 = 0.96006944477
0.9184027781 + -1.916666667x + x2 = 0.96006944477

Factor a perfect square on the left side:
(x + -0.9583333335)(x + -0.9583333335) = 0.96006944477

Calculate the square root of the right side: 0.979831335

Break this problem into two subproblems by setting 
(x + -0.9583333335) equal to 0.979831335 and -0.979831335.

Subproblem 1

x + -0.9583333335 = 0.979831335 Simplifying x + -0.9583333335 = 0.979831335 Reorder the terms: -0.9583333335 + x = 0.979831335 Solving -0.9583333335 + x = 0.979831335 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9583333335' to each side of the equation. -0.9583333335 + 0.9583333335 + x = 0.979831335 + 0.9583333335 Combine like terms: -0.9583333335 + 0.9583333335 = 0.0000000000 0.0000000000 + x = 0.979831335 + 0.9583333335 x = 0.979831335 + 0.9583333335 Combine like terms: 0.979831335 + 0.9583333335 = 1.9381646685 x = 1.9381646685 Simplifying x = 1.9381646685

Subproblem 2

x + -0.9583333335 = -0.979831335 Simplifying x + -0.9583333335 = -0.979831335 Reorder the terms: -0.9583333335 + x = -0.979831335 Solving -0.9583333335 + x = -0.979831335 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9583333335' to each side of the equation. -0.9583333335 + 0.9583333335 + x = -0.979831335 + 0.9583333335 Combine like terms: -0.9583333335 + 0.9583333335 = 0.0000000000 0.0000000000 + x = -0.979831335 + 0.9583333335 x = -0.979831335 + 0.9583333335 Combine like terms: -0.979831335 + 0.9583333335 = -0.0214980015 x = -0.0214980015 Simplifying x = -0.0214980015

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.9381646685, -0.0214980015}

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