16x^2-24x-23=0

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Solution for 16x^2-24x-23=0 equation:


Simplifying
16x2 + -24x + -23 = 0

Reorder the terms:
-23 + -24x + 16x2 = 0

Solving
-23 + -24x + 16x2 = 0

Solving for variable 'x'.

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

Divide each side by '16'.
-1.4375 + -1.5x + x2 = 0

Move the constant term to the right:

Add '1.4375' to each side of the equation.
-1.4375 + -1.5x + 1.4375 + x2 = 0 + 1.4375

Reorder the terms:
-1.4375 + 1.4375 + -1.5x + x2 = 0 + 1.4375

Combine like terms: -1.4375 + 1.4375 = 0.0000
0.0000 + -1.5x + x2 = 0 + 1.4375
-1.5x + x2 = 0 + 1.4375

Combine like terms: 0 + 1.4375 = 1.4375
-1.5x + x2 = 1.4375

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 1.4375 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 1.4375 + 0.5625

Combine like terms: 1.4375 + 0.5625 = 2
0.5625 + -1.5x + x2 = 2

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + -0.75) equal to 1.414213562 and -1.414213562.

Subproblem 1

x + -0.75 = 1.414213562 Simplifying x + -0.75 = 1.414213562 Reorder the terms: -0.75 + x = 1.414213562 Solving -0.75 + x = 1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 1.414213562 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 1.414213562 + 0.75 x = 1.414213562 + 0.75 Combine like terms: 1.414213562 + 0.75 = 2.164213562 x = 2.164213562 Simplifying x = 2.164213562

Subproblem 2

x + -0.75 = -1.414213562 Simplifying x + -0.75 = -1.414213562 Reorder the terms: -0.75 + x = -1.414213562 Solving -0.75 + x = -1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -1.414213562 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -1.414213562 + 0.75 x = -1.414213562 + 0.75 Combine like terms: -1.414213562 + 0.75 = -0.664213562 x = -0.664213562 Simplifying x = -0.664213562

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.164213562, -0.664213562}

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