4x^2+2x-23=0

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


Simplifying
4x2 + 2x + -23 = 0

Reorder the terms:
-23 + 2x + 4x2 = 0

Solving
-23 + 2x + 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 + 0.5x + x2 = 0

Move the constant term to the right:

Add '5.75' to each side of the equation.
-5.75 + 0.5x + 5.75 + x2 = 0 + 5.75

Reorder the terms:
-5.75 + 5.75 + 0.5x + x2 = 0 + 5.75

Combine like terms: -5.75 + 5.75 = 0.00
0.00 + 0.5x + x2 = 0 + 5.75
0.5x + x2 = 0 + 5.75

Combine like terms: 0 + 5.75 = 5.75
0.5x + x2 = 5.75

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 5.75 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 5.75 + 0.0625

Combine like terms: 5.75 + 0.0625 = 5.8125
0.0625 + 0.5x + x2 = 5.8125

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 5.8125

Calculate the square root of the right side: 2.41091269

Break this problem into two subproblems by setting 
(x + 0.25) equal to 2.41091269 and -2.41091269.

Subproblem 1

x + 0.25 = 2.41091269 Simplifying x + 0.25 = 2.41091269 Reorder the terms: 0.25 + x = 2.41091269 Solving 0.25 + x = 2.41091269 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 2.41091269 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 2.41091269 + -0.25 x = 2.41091269 + -0.25 Combine like terms: 2.41091269 + -0.25 = 2.16091269 x = 2.16091269 Simplifying x = 2.16091269

Subproblem 2

x + 0.25 = -2.41091269 Simplifying x + 0.25 = -2.41091269 Reorder the terms: 0.25 + x = -2.41091269 Solving 0.25 + x = -2.41091269 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -2.41091269 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -2.41091269 + -0.25 x = -2.41091269 + -0.25 Combine like terms: -2.41091269 + -0.25 = -2.66091269 x = -2.66091269 Simplifying x = -2.66091269

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.16091269, -2.66091269}

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