4x^2-23x-16=0

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Solution for 4x^2-23x-16=0 equation:


Simplifying
4x2 + -23x + -16 = 0

Reorder the terms:
-16 + -23x + 4x2 = 0

Solving
-16 + -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'.
-4 + -5.75x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -5.75x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + -5.75x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -5.75x + x2 = 0 + 4
-5.75x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
-5.75x + x2 = 4

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 = 4 + 8.265625

Reorder the terms:
8.265625 + -5.75x + x2 = 4 + 8.265625

Combine like terms: 4 + 8.265625 = 12.265625
8.265625 + -5.75x + x2 = 12.265625

Factor a perfect square on the left side:
(x + -2.875)(x + -2.875) = 12.265625

Calculate the square root of the right side: 3.502231432

Break this problem into two subproblems by setting 
(x + -2.875) equal to 3.502231432 and -3.502231432.

Subproblem 1

x + -2.875 = 3.502231432 Simplifying x + -2.875 = 3.502231432 Reorder the terms: -2.875 + x = 3.502231432 Solving -2.875 + x = 3.502231432 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 = 3.502231432 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + x = 3.502231432 + 2.875 x = 3.502231432 + 2.875 Combine like terms: 3.502231432 + 2.875 = 6.377231432 x = 6.377231432 Simplifying x = 6.377231432

Subproblem 2

x + -2.875 = -3.502231432 Simplifying x + -2.875 = -3.502231432 Reorder the terms: -2.875 + x = -3.502231432 Solving -2.875 + x = -3.502231432 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 = -3.502231432 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + x = -3.502231432 + 2.875 x = -3.502231432 + 2.875 Combine like terms: -3.502231432 + 2.875 = -0.627231432 x = -0.627231432 Simplifying x = -0.627231432

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.377231432, -0.627231432}

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