7x^2+23x-24=0

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Solution for 7x^2+23x-24=0 equation: 


Simplifying
7x2 + 23x + -24 = 0

Reorder the terms:
-24 + 23x + 7x2 = 0

Solving
-24 + 23x + 7x2 = 0

Solving for variable 'x'.

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

Divide each side by '7'.
-3.428571429 + 3.285714286x + x2 = 0

Move the constant term to the right:

Add '3.428571429' to each side of the equation.
-3.428571429 + 3.285714286x + 3.428571429 + x2 = 0 + 3.428571429

Reorder the terms:
-3.428571429 + 3.428571429 + 3.285714286x + x2 = 0 + 3.428571429

Combine like terms: -3.428571429 + 3.428571429 = 0.000000000
0.000000000 + 3.285714286x + x2 = 0 + 3.428571429
3.285714286x + x2 = 0 + 3.428571429

Combine like terms: 0 + 3.428571429 = 3.428571429
3.285714286x + x2 = 3.428571429

The x term is 3.285714286x.  Take half its coefficient (1.642857143).
Square it (2.698979592) and add it to both sides.

Add '2.698979592' to each side of the equation.
3.285714286x + 2.698979592 + x2 = 3.428571429 + 2.698979592

Reorder the terms:
2.698979592 + 3.285714286x + x2 = 3.428571429 + 2.698979592

Combine like terms: 3.428571429 + 2.698979592 = 6.127551021
2.698979592 + 3.285714286x + x2 = 6.127551021

Factor a perfect square on the left side:
(x + 1.642857143)(x + 1.642857143) = 6.127551021

Calculate the square root of the right side: 2.475389065

Break this problem into two subproblems by setting 
(x + 1.642857143) equal to 2.475389065 and -2.475389065.


Subproblem 1
x + 1.642857143 = 2.475389065

Simplifying
x + 1.642857143 = 2.475389065

Reorder the terms:
1.642857143 + x = 2.475389065

Solving
1.642857143 + x = 2.475389065

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.642857143' to each side of the equation.
1.642857143 + -1.642857143 + x = 2.475389065 + -1.642857143

Combine like terms: 1.642857143 + -1.642857143 = 0.000000000
0.000000000 + x = 2.475389065 + -1.642857143
x = 2.475389065 + -1.642857143

Combine like terms: 2.475389065 + -1.642857143 = 0.832531922
x = 0.832531922

Simplifying
x = 0.832531922


Subproblem 2
x + 1.642857143 = -2.475389065

Simplifying
x + 1.642857143 = -2.475389065

Reorder the terms:
1.642857143 + x = -2.475389065

Solving
1.642857143 + x = -2.475389065

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.642857143' to each side of the equation.
1.642857143 + -1.642857143 + x = -2.475389065 + -1.642857143

Combine like terms: 1.642857143 + -1.642857143 = 0.000000000
0.000000000 + x = -2.475389065 + -1.642857143
x = -2.475389065 + -1.642857143

Combine like terms: -2.475389065 + -1.642857143 = -4.118246208
x = -4.118246208

Simplifying
x = -4.118246208


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.832531922, -4.118246208}

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