24x^2+23x-24=0

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Solution for 24x^2+23x-24=0 equation:


Simplifying
24x2 + 23x + -24 = 0

Reorder the terms:
-24 + 23x + 24x2 = 0

Solving
-24 + 23x + 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'.
-1 + 0.9583333333x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.9583333333x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.9583333333x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.9583333333x + x2 = 0 + 1
0.9583333333x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.9583333333x + x2 = 1

The x term is 0.9583333333x.  Take half its coefficient (0.4791666667).
Square it (0.2296006945) and add it to both sides.

Add '0.2296006945' to each side of the equation.
0.9583333333x + 0.2296006945 + x2 = 1 + 0.2296006945

Reorder the terms:
0.2296006945 + 0.9583333333x + x2 = 1 + 0.2296006945

Combine like terms: 1 + 0.2296006945 = 1.2296006945
0.2296006945 + 0.9583333333x + x2 = 1.2296006945

Factor a perfect square on the left side:
(x + 0.4791666667)(x + 0.4791666667) = 1.2296006945

Calculate the square root of the right side: 1.108873615

Break this problem into two subproblems by setting 
(x + 0.4791666667) equal to 1.108873615 and -1.108873615.

Subproblem 1

x + 0.4791666667 = 1.108873615 Simplifying x + 0.4791666667 = 1.108873615 Reorder the terms: 0.4791666667 + x = 1.108873615 Solving 0.4791666667 + x = 1.108873615 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4791666667' to each side of the equation. 0.4791666667 + -0.4791666667 + x = 1.108873615 + -0.4791666667 Combine like terms: 0.4791666667 + -0.4791666667 = 0.0000000000 0.0000000000 + x = 1.108873615 + -0.4791666667 x = 1.108873615 + -0.4791666667 Combine like terms: 1.108873615 + -0.4791666667 = 0.6297069483 x = 0.6297069483 Simplifying x = 0.6297069483

Subproblem 2

x + 0.4791666667 = -1.108873615 Simplifying x + 0.4791666667 = -1.108873615 Reorder the terms: 0.4791666667 + x = -1.108873615 Solving 0.4791666667 + x = -1.108873615 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4791666667' to each side of the equation. 0.4791666667 + -0.4791666667 + x = -1.108873615 + -0.4791666667 Combine like terms: 0.4791666667 + -0.4791666667 = 0.0000000000 0.0000000000 + x = -1.108873615 + -0.4791666667 x = -1.108873615 + -0.4791666667 Combine like terms: -1.108873615 + -0.4791666667 = -1.5880402817 x = -1.5880402817 Simplifying x = -1.5880402817

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.6297069483, -1.5880402817}

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