12x^2-5x-24=0

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Solution for 12x^2-5x-24=0 equation:


Simplifying
12x2 + -5x + -24 = 0

Reorder the terms:
-24 + -5x + 12x2 = 0

Solving
-24 + -5x + 12x2 = 0

Solving for variable 'x'.

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

Divide each side by '12'.
-2 + -0.4166666667x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -0.4166666667x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -0.4166666667x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -0.4166666667x + x2 = 0 + 2
-0.4166666667x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-0.4166666667x + x2 = 2

The x term is -0.4166666667x.  Take half its coefficient (-0.2083333334).
Square it (0.04340277781) and add it to both sides.

Add '0.04340277781' to each side of the equation.
-0.4166666667x + 0.04340277781 + x2 = 2 + 0.04340277781

Reorder the terms:
0.04340277781 + -0.4166666667x + x2 = 2 + 0.04340277781

Combine like terms: 2 + 0.04340277781 = 2.04340277781
0.04340277781 + -0.4166666667x + x2 = 2.04340277781

Factor a perfect square on the left side:
(x + -0.2083333334)(x + -0.2083333334) = 2.04340277781

Calculate the square root of the right side: 1.4294764

Break this problem into two subproblems by setting 
(x + -0.2083333334) equal to 1.4294764 and -1.4294764.

Subproblem 1

x + -0.2083333334 = 1.4294764 Simplifying x + -0.2083333334 = 1.4294764 Reorder the terms: -0.2083333334 + x = 1.4294764 Solving -0.2083333334 + x = 1.4294764 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2083333334' to each side of the equation. -0.2083333334 + 0.2083333334 + x = 1.4294764 + 0.2083333334 Combine like terms: -0.2083333334 + 0.2083333334 = 0.0000000000 0.0000000000 + x = 1.4294764 + 0.2083333334 x = 1.4294764 + 0.2083333334 Combine like terms: 1.4294764 + 0.2083333334 = 1.6378097334 x = 1.6378097334 Simplifying x = 1.6378097334

Subproblem 2

x + -0.2083333334 = -1.4294764 Simplifying x + -0.2083333334 = -1.4294764 Reorder the terms: -0.2083333334 + x = -1.4294764 Solving -0.2083333334 + x = -1.4294764 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2083333334' to each side of the equation. -0.2083333334 + 0.2083333334 + x = -1.4294764 + 0.2083333334 Combine like terms: -0.2083333334 + 0.2083333334 = 0.0000000000 0.0000000000 + x = -1.4294764 + 0.2083333334 x = -1.4294764 + 0.2083333334 Combine like terms: -1.4294764 + 0.2083333334 = -1.2211430666 x = -1.2211430666 Simplifying x = -1.2211430666

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.6378097334, -1.2211430666}

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