24x^2+5x-10=0

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Solution for 24x^2+5x-10=0 equation:


Simplifying
24x2 + 5x + -10 = 0

Reorder the terms:
-10 + 5x + 24x2 = 0

Solving
-10 + 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'.
-0.4166666667 + 0.2083333333x + x2 = 0

Move the constant term to the right:

Add '0.4166666667' to each side of the equation.
-0.4166666667 + 0.2083333333x + 0.4166666667 + x2 = 0 + 0.4166666667

Reorder the terms:
-0.4166666667 + 0.4166666667 + 0.2083333333x + x2 = 0 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + 0.2083333333x + x2 = 0 + 0.4166666667
0.2083333333x + x2 = 0 + 0.4166666667

Combine like terms: 0 + 0.4166666667 = 0.4166666667
0.2083333333x + x2 = 0.4166666667

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 = 0.4166666667 + 0.01085069445

Reorder the terms:
0.01085069445 + 0.2083333333x + x2 = 0.4166666667 + 0.01085069445

Combine like terms: 0.4166666667 + 0.01085069445 = 0.42751736115
0.01085069445 + 0.2083333333x + x2 = 0.42751736115

Factor a perfect square on the left side:
(x + 0.1041666667)(x + 0.1041666667) = 0.42751736115

Calculate the square root of the right side: 0.653848118

Break this problem into two subproblems by setting 
(x + 0.1041666667) equal to 0.653848118 and -0.653848118.


Subproblem 1
x + 0.1041666667 = 0.653848118

Simplifying
x + 0.1041666667 = 0.653848118

Reorder the terms:
0.1041666667 + x = 0.653848118

Solving
0.1041666667 + x = 0.653848118

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 = 0.653848118 + -0.1041666667

Combine like terms: 0.1041666667 + -0.1041666667 = 0.0000000000
0.0000000000 + x = 0.653848118 + -0.1041666667
x = 0.653848118 + -0.1041666667

Combine like terms: 0.653848118 + -0.1041666667 = 0.5496814513
x = 0.5496814513

Simplifying
x = 0.5496814513


Subproblem 2
x + 0.1041666667 = -0.653848118

Simplifying
x + 0.1041666667 = -0.653848118

Reorder the terms:
0.1041666667 + x = -0.653848118

Solving
0.1041666667 + x = -0.653848118

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 = -0.653848118 + -0.1041666667

Combine like terms: 0.1041666667 + -0.1041666667 = 0.0000000000
0.0000000000 + x = -0.653848118 + -0.1041666667
x = -0.653848118 + -0.1041666667

Combine like terms: -0.653848118 + -0.1041666667 = -0.7580147847
x = -0.7580147847

Simplifying
x = -0.7580147847


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.5496814513, -0.7580147847}

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