4x+5=x^2+5x-24

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Solution for 4x+5=x^2+5x-24 equation:


Simplifying
4x + 5 = x2 + 5x + -24

Reorder the terms:
5 + 4x = x2 + 5x + -24

Reorder the terms:
5 + 4x = -24 + 5x + x2

Solving
5 + 4x = -24 + 5x + x2

Solving for variable 'x'.

Reorder the terms:
5 + 24 + 4x + -5x + -1x2 = -24 + 5x + x2 + 24 + -5x + -1x2

Combine like terms: 5 + 24 = 29
29 + 4x + -5x + -1x2 = -24 + 5x + x2 + 24 + -5x + -1x2

Combine like terms: 4x + -5x = -1x
29 + -1x + -1x2 = -24 + 5x + x2 + 24 + -5x + -1x2

Reorder the terms:
29 + -1x + -1x2 = -24 + 24 + 5x + -5x + x2 + -1x2

Combine like terms: -24 + 24 = 0
29 + -1x + -1x2 = 0 + 5x + -5x + x2 + -1x2
29 + -1x + -1x2 = 5x + -5x + x2 + -1x2

Combine like terms: 5x + -5x = 0
29 + -1x + -1x2 = 0 + x2 + -1x2
29 + -1x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
29 + -1x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-29 + x + x2 = 0

Move the constant term to the right:

Add '29' to each side of the equation.
-29 + x + 29 + x2 = 0 + 29

Reorder the terms:
-29 + 29 + x + x2 = 0 + 29

Combine like terms: -29 + 29 = 0
0 + x + x2 = 0 + 29
x + x2 = 0 + 29

Combine like terms: 0 + 29 = 29
x + x2 = 29

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

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 29 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 29 + 0.25

Combine like terms: 29 + 0.25 = 29.25
1.25 + x2 = 29.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 29.25

Calculate the square root of the right side: 5.408326913

Break this problem into two subproblems by setting 
(x + 0.5) equal to 5.408326913 and -5.408326913.

Subproblem 1

x + 0.5 = 5.408326913 Simplifying x + 0.5 = 5.408326913 Reorder the terms: 0.5 + x = 5.408326913 Solving 0.5 + x = 5.408326913 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 5.408326913 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 5.408326913 + -0.5 x = 5.408326913 + -0.5 Combine like terms: 5.408326913 + -0.5 = 4.908326913 x = 4.908326913 Simplifying x = 4.908326913

Subproblem 2

x + 0.5 = -5.408326913 Simplifying x + 0.5 = -5.408326913 Reorder the terms: 0.5 + x = -5.408326913 Solving 0.5 + x = -5.408326913 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -5.408326913 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -5.408326913 + -0.5 x = -5.408326913 + -0.5 Combine like terms: -5.408326913 + -0.5 = -5.908326913 x = -5.908326913 Simplifying x = -5.908326913

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.908326913, -5.908326913}

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