24x^2-5x-60=0

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Solution for 24x^2-5x-60=0 equation:


Simplifying
24x2 + -5x + -60 = 0

Reorder the terms:
-60 + -5x + 24x2 = 0

Solving
-60 + -5x + 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'.
-2.5 + -0.2083333333x + x2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -0.2083333333x + 2.5 + x2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -0.2083333333x + x2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -0.2083333333x + x2 = 0 + 2.5
-0.2083333333x + x2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-0.2083333333x + x2 = 2.5

The x term is -0.2083333333x.  Take half its coefficient (-0.1041666667).
Square it (0.01085069445) and add it to both sides.

Add '0.01085069445' to each side of the equation.
-0.2083333333x + 0.01085069445 + x2 = 2.5 + 0.01085069445

Reorder the terms:
0.01085069445 + -0.2083333333x + x2 = 2.5 + 0.01085069445

Combine like terms: 2.5 + 0.01085069445 = 2.51085069445
0.01085069445 + -0.2083333333x + x2 = 2.51085069445

Factor a perfect square on the left side:
(x + -0.1041666667)(x + -0.1041666667) = 2.51085069445

Calculate the square root of the right side: 1.584566406

Break this problem into two subproblems by setting 
(x + -0.1041666667) equal to 1.584566406 and -1.584566406.

Subproblem 1

x + -0.1041666667 = 1.584566406 Simplifying x + -0.1041666667 = 1.584566406 Reorder the terms: -0.1041666667 + x = 1.584566406 Solving -0.1041666667 + x = 1.584566406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1041666667' to each side of the equation. -0.1041666667 + 0.1041666667 + x = 1.584566406 + 0.1041666667 Combine like terms: -0.1041666667 + 0.1041666667 = 0.0000000000 0.0000000000 + x = 1.584566406 + 0.1041666667 x = 1.584566406 + 0.1041666667 Combine like terms: 1.584566406 + 0.1041666667 = 1.6887330727 x = 1.6887330727 Simplifying x = 1.6887330727

Subproblem 2

x + -0.1041666667 = -1.584566406 Simplifying x + -0.1041666667 = -1.584566406 Reorder the terms: -0.1041666667 + x = -1.584566406 Solving -0.1041666667 + x = -1.584566406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1041666667' to each side of the equation. -0.1041666667 + 0.1041666667 + x = -1.584566406 + 0.1041666667 Combine like terms: -0.1041666667 + 0.1041666667 = 0.0000000000 0.0000000000 + x = -1.584566406 + 0.1041666667 x = -1.584566406 + 0.1041666667 Combine like terms: -1.584566406 + 0.1041666667 = -1.4803997393 x = -1.4803997393 Simplifying x = -1.4803997393

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.6887330727, -1.4803997393}

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