48x^2-73x+23=0

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Solution for 48x^2-73x+23=0 equation:


Simplifying
48x2 + -73x + 23 = 0

Reorder the terms:
23 + -73x + 48x2 = 0

Solving
23 + -73x + 48x2 = 0

Solving for variable 'x'.

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

Divide each side by '48'.
0.4791666667 + -1.520833333x + x2 = 0

Move the constant term to the right:

Add '-0.4791666667' to each side of the equation.
0.4791666667 + -1.520833333x + -0.4791666667 + x2 = 0 + -0.4791666667

Reorder the terms:
0.4791666667 + -0.4791666667 + -1.520833333x + x2 = 0 + -0.4791666667

Combine like terms: 0.4791666667 + -0.4791666667 = 0.0000000000
0.0000000000 + -1.520833333x + x2 = 0 + -0.4791666667
-1.520833333x + x2 = 0 + -0.4791666667

Combine like terms: 0 + -0.4791666667 = -0.4791666667
-1.520833333x + x2 = -0.4791666667

The x term is -1.520833333x.  Take half its coefficient (-0.7604166665).
Square it (0.5782335067) and add it to both sides.

Add '0.5782335067' to each side of the equation.
-1.520833333x + 0.5782335067 + x2 = -0.4791666667 + 0.5782335067

Reorder the terms:
0.5782335067 + -1.520833333x + x2 = -0.4791666667 + 0.5782335067

Combine like terms: -0.4791666667 + 0.5782335067 = 0.09906684
0.5782335067 + -1.520833333x + x2 = 0.09906684

Factor a perfect square on the left side:
(x + -0.7604166665)(x + -0.7604166665) = 0.09906684

Calculate the square root of the right side: 0.314748852

Break this problem into two subproblems by setting 
(x + -0.7604166665) equal to 0.314748852 and -0.314748852.

Subproblem 1

x + -0.7604166665 = 0.314748852 Simplifying x + -0.7604166665 = 0.314748852 Reorder the terms: -0.7604166665 + x = 0.314748852 Solving -0.7604166665 + x = 0.314748852 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.7604166665' to each side of the equation. -0.7604166665 + 0.7604166665 + x = 0.314748852 + 0.7604166665 Combine like terms: -0.7604166665 + 0.7604166665 = 0.0000000000 0.0000000000 + x = 0.314748852 + 0.7604166665 x = 0.314748852 + 0.7604166665 Combine like terms: 0.314748852 + 0.7604166665 = 1.0751655185 x = 1.0751655185 Simplifying x = 1.0751655185

Subproblem 2

x + -0.7604166665 = -0.314748852 Simplifying x + -0.7604166665 = -0.314748852 Reorder the terms: -0.7604166665 + x = -0.314748852 Solving -0.7604166665 + x = -0.314748852 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.7604166665' to each side of the equation. -0.7604166665 + 0.7604166665 + x = -0.314748852 + 0.7604166665 Combine like terms: -0.7604166665 + 0.7604166665 = 0.0000000000 0.0000000000 + x = -0.314748852 + 0.7604166665 x = -0.314748852 + 0.7604166665 Combine like terms: -0.314748852 + 0.7604166665 = 0.4456678145 x = 0.4456678145 Simplifying x = 0.4456678145

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.0751655185, 0.4456678145}

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