4x^2+8x-23=0

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Solution for 4x^2+8x-23=0 equation:


Simplifying
4x2 + 8x + -23 = 0

Reorder the terms:
-23 + 8x + 4x2 = 0

Solving
-23 + 8x + 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 + 2x + x2 = 0

Move the constant term to the right:

Add '5.75' to each side of the equation.
-5.75 + 2x + 5.75 + x2 = 0 + 5.75

Reorder the terms:
-5.75 + 5.75 + 2x + x2 = 0 + 5.75

Combine like terms: -5.75 + 5.75 = 0.00
0.00 + 2x + x2 = 0 + 5.75
2x + x2 = 0 + 5.75

Combine like terms: 0 + 5.75 = 5.75
2x + x2 = 5.75

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 5.75 + 1

Reorder the terms:
1 + 2x + x2 = 5.75 + 1

Combine like terms: 5.75 + 1 = 6.75
1 + 2x + x2 = 6.75

Factor a perfect square on the left side:
(x + 1)(x + 1) = 6.75

Calculate the square root of the right side: 2.598076211

Break this problem into two subproblems by setting 
(x + 1) equal to 2.598076211 and -2.598076211.

Subproblem 1

x + 1 = 2.598076211 Simplifying x + 1 = 2.598076211 Reorder the terms: 1 + x = 2.598076211 Solving 1 + x = 2.598076211 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 2.598076211 + -1 Combine like terms: 1 + -1 = 0 0 + x = 2.598076211 + -1 x = 2.598076211 + -1 Combine like terms: 2.598076211 + -1 = 1.598076211 x = 1.598076211 Simplifying x = 1.598076211

Subproblem 2

x + 1 = -2.598076211 Simplifying x + 1 = -2.598076211 Reorder the terms: 1 + x = -2.598076211 Solving 1 + x = -2.598076211 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -2.598076211 + -1 Combine like terms: 1 + -1 = 0 0 + x = -2.598076211 + -1 x = -2.598076211 + -1 Combine like terms: -2.598076211 + -1 = -3.598076211 x = -3.598076211 Simplifying x = -3.598076211

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.598076211, -3.598076211}

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