4x^2-21x-23=0

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Solution for 4x^2-21x-23=0 equation:


Simplifying
4x2 + -21x + -23 = 0

Reorder the terms:
-23 + -21x + 4x2 = 0

Solving
-23 + -21x + 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'.
-5.75 + -5.25x + x2 = 0

Move the constant term to the right:

Add '5.75' to each side of the equation.
-5.75 + -5.25x + 5.75 + x2 = 0 + 5.75

Reorder the terms:
-5.75 + 5.75 + -5.25x + x2 = 0 + 5.75

Combine like terms: -5.75 + 5.75 = 0.00
0.00 + -5.25x + x2 = 0 + 5.75
-5.25x + x2 = 0 + 5.75

Combine like terms: 0 + 5.75 = 5.75
-5.25x + x2 = 5.75

The x term is -5.25x.  Take half its coefficient (-2.625).
Square it (6.890625) and add it to both sides.

Add '6.890625' to each side of the equation.
-5.25x + 6.890625 + x2 = 5.75 + 6.890625

Reorder the terms:
6.890625 + -5.25x + x2 = 5.75 + 6.890625

Combine like terms: 5.75 + 6.890625 = 12.640625
6.890625 + -5.25x + x2 = 12.640625

Factor a perfect square on the left side:
(x + -2.625)(x + -2.625) = 12.640625

Calculate the square root of the right side: 3.555365663

Break this problem into two subproblems by setting 
(x + -2.625) equal to 3.555365663 and -3.555365663.

Subproblem 1

x + -2.625 = 3.555365663 Simplifying x + -2.625 = 3.555365663 Reorder the terms: -2.625 + x = 3.555365663 Solving -2.625 + x = 3.555365663 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + x = 3.555365663 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + x = 3.555365663 + 2.625 x = 3.555365663 + 2.625 Combine like terms: 3.555365663 + 2.625 = 6.180365663 x = 6.180365663 Simplifying x = 6.180365663

Subproblem 2

x + -2.625 = -3.555365663 Simplifying x + -2.625 = -3.555365663 Reorder the terms: -2.625 + x = -3.555365663 Solving -2.625 + x = -3.555365663 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + x = -3.555365663 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + x = -3.555365663 + 2.625 x = -3.555365663 + 2.625 Combine like terms: -3.555365663 + 2.625 = -0.930365663 x = -0.930365663 Simplifying x = -0.930365663

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.180365663, -0.930365663}

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