4x^2+23x=-24

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Solution for 4x^2+23x=-24 equation:


Simplifying
4x2 + 23x = -24

Reorder the terms:
23x + 4x2 = -24

Solving
23x + 4x2 = -24

Solving for variable 'x'.

Reorder the terms:
24 + 23x + 4x2 = -24 + 24

Combine like terms: -24 + 24 = 0
24 + 23x + 4x2 = 0

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

Divide each side by '4'.
6 + 5.75x + x2 = 0

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + 5.75x + -6 + x2 = 0 + -6

Reorder the terms:
6 + -6 + 5.75x + x2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + 5.75x + x2 = 0 + -6
5.75x + x2 = 0 + -6

Combine like terms: 0 + -6 = -6
5.75x + x2 = -6

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 = -6 + 8.265625

Reorder the terms:
8.265625 + 5.75x + x2 = -6 + 8.265625

Combine like terms: -6 + 8.265625 = 2.265625
8.265625 + 5.75x + x2 = 2.265625

Factor a perfect square on the left side:
(x + 2.875)(x + 2.875) = 2.265625

Calculate the square root of the right side: 1.505199322

Break this problem into two subproblems by setting 
(x + 2.875) equal to 1.505199322 and -1.505199322.

Subproblem 1

x + 2.875 = 1.505199322 Simplifying x + 2.875 = 1.505199322 Reorder the terms: 2.875 + x = 1.505199322 Solving 2.875 + x = 1.505199322 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 = 1.505199322 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + x = 1.505199322 + -2.875 x = 1.505199322 + -2.875 Combine like terms: 1.505199322 + -2.875 = -1.369800678 x = -1.369800678 Simplifying x = -1.369800678

Subproblem 2

x + 2.875 = -1.505199322 Simplifying x + 2.875 = -1.505199322 Reorder the terms: 2.875 + x = -1.505199322 Solving 2.875 + x = -1.505199322 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 = -1.505199322 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + x = -1.505199322 + -2.875 x = -1.505199322 + -2.875 Combine like terms: -1.505199322 + -2.875 = -4.380199322 x = -4.380199322 Simplifying x = -4.380199322

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.369800678, -4.380199322}

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