24x^2-19x-11=0

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Solution for 24x^2-19x-11=0 equation:


Simplifying
24x2 + -19x + -11 = 0

Reorder the terms:
-11 + -19x + 24x2 = 0

Solving
-11 + -19x + 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'.
-0.4583333333 + -0.7916666667x + x2 = 0

Move the constant term to the right:

Add '0.4583333333' to each side of the equation.
-0.4583333333 + -0.7916666667x + 0.4583333333 + x2 = 0 + 0.4583333333

Reorder the terms:
-0.4583333333 + 0.4583333333 + -0.7916666667x + x2 = 0 + 0.4583333333

Combine like terms: -0.4583333333 + 0.4583333333 = 0.0000000000
0.0000000000 + -0.7916666667x + x2 = 0 + 0.4583333333
-0.7916666667x + x2 = 0 + 0.4583333333

Combine like terms: 0 + 0.4583333333 = 0.4583333333
-0.7916666667x + x2 = 0.4583333333

The x term is -0.7916666667x.  Take half its coefficient (-0.3958333334).
Square it (0.1566840278) and add it to both sides.

Add '0.1566840278' to each side of the equation.
-0.7916666667x + 0.1566840278 + x2 = 0.4583333333 + 0.1566840278

Reorder the terms:
0.1566840278 + -0.7916666667x + x2 = 0.4583333333 + 0.1566840278

Combine like terms: 0.4583333333 + 0.1566840278 = 0.6150173611
0.1566840278 + -0.7916666667x + x2 = 0.6150173611

Factor a perfect square on the left side:
(x + -0.3958333334)(x + -0.3958333334) = 0.6150173611

Calculate the square root of the right side: 0.784230426

Break this problem into two subproblems by setting 
(x + -0.3958333334) equal to 0.784230426 and -0.784230426.

Subproblem 1

x + -0.3958333334 = 0.784230426 Simplifying x + -0.3958333334 = 0.784230426 Reorder the terms: -0.3958333334 + x = 0.784230426 Solving -0.3958333334 + x = 0.784230426 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3958333334' to each side of the equation. -0.3958333334 + 0.3958333334 + x = 0.784230426 + 0.3958333334 Combine like terms: -0.3958333334 + 0.3958333334 = 0.0000000000 0.0000000000 + x = 0.784230426 + 0.3958333334 x = 0.784230426 + 0.3958333334 Combine like terms: 0.784230426 + 0.3958333334 = 1.1800637594 x = 1.1800637594 Simplifying x = 1.1800637594

Subproblem 2

x + -0.3958333334 = -0.784230426 Simplifying x + -0.3958333334 = -0.784230426 Reorder the terms: -0.3958333334 + x = -0.784230426 Solving -0.3958333334 + x = -0.784230426 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3958333334' to each side of the equation. -0.3958333334 + 0.3958333334 + x = -0.784230426 + 0.3958333334 Combine like terms: -0.3958333334 + 0.3958333334 = 0.0000000000 0.0000000000 + x = -0.784230426 + 0.3958333334 x = -0.784230426 + 0.3958333334 Combine like terms: -0.784230426 + 0.3958333334 = -0.3883970926 x = -0.3883970926 Simplifying x = -0.3883970926

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.1800637594, -0.3883970926}

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