4x^2-23x+9=0

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Solution for 4x^2-23x+9=0 equation:


Simplifying
4x2 + -23x + 9 = 0

Reorder the terms:
9 + -23x + 4x2 = 0

Solving
9 + -23x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
2.25 + -5.75x + x2 = 0

Move the constant term to the right:

Add '-2.25' to each side of the equation.
2.25 + -5.75x + -2.25 + x2 = 0 + -2.25

Reorder the terms:
2.25 + -2.25 + -5.75x + x2 = 0 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + -5.75x + x2 = 0 + -2.25
-5.75x + x2 = 0 + -2.25

Combine like terms: 0 + -2.25 = -2.25
-5.75x + x2 = -2.25

The x term is -5.75x.  Take half its coefficient (-2.875).
Square it (8.265625) and add it to both sides.

Add '8.265625' to each side of the equation.
-5.75x + 8.265625 + x2 = -2.25 + 8.265625

Reorder the terms:
8.265625 + -5.75x + x2 = -2.25 + 8.265625

Combine like terms: -2.25 + 8.265625 = 6.015625
8.265625 + -5.75x + x2 = 6.015625

Factor a perfect square on the left side:
(x + -2.875)(x + -2.875) = 6.015625

Calculate the square root of the right side: 2.452677109

Break this problem into two subproblems by setting 
(x + -2.875) equal to 2.452677109 and -2.452677109.

Subproblem 1

x + -2.875 = 2.452677109 Simplifying x + -2.875 = 2.452677109 Reorder the terms: -2.875 + x = 2.452677109 Solving -2.875 + x = 2.452677109 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.875' to each side of the equation. -2.875 + 2.875 + x = 2.452677109 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + x = 2.452677109 + 2.875 x = 2.452677109 + 2.875 Combine like terms: 2.452677109 + 2.875 = 5.327677109 x = 5.327677109 Simplifying x = 5.327677109

Subproblem 2

x + -2.875 = -2.452677109 Simplifying x + -2.875 = -2.452677109 Reorder the terms: -2.875 + x = -2.452677109 Solving -2.875 + x = -2.452677109 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.875' to each side of the equation. -2.875 + 2.875 + x = -2.452677109 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + x = -2.452677109 + 2.875 x = -2.452677109 + 2.875 Combine like terms: -2.452677109 + 2.875 = 0.422322891 x = 0.422322891 Simplifying x = 0.422322891

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.327677109, 0.422322891}

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