3x^2+7x-6=24

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Solution for 3x^2+7x-6=24 equation: 


Simplifying
3x2 + 7x + -6 = 24

Reorder the terms:
-6 + 7x + 3x2 = 24

Solving
-6 + 7x + 3x2 = 24

Solving for variable 'x'.

Reorder the terms:
-6 + -24 + 7x + 3x2 = 24 + -24

Combine like terms: -6 + -24 = -30
-30 + 7x + 3x2 = 24 + -24

Combine like terms: 24 + -24 = 0
-30 + 7x + 3x2 = 0

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

Divide each side by '3'.
-10 + 2.333333333x + x2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 2.333333333x + 10 + x2 = 0 + 10

Reorder the terms:
-10 + 10 + 2.333333333x + x2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 2.333333333x + x2 = 0 + 10
2.333333333x + x2 = 0 + 10

Combine like terms: 0 + 10 = 10
2.333333333x + x2 = 10

The x term is 2.333333333x.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333x + 1.361111112 + x2 = 10 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333x + x2 = 10 + 1.361111112

Combine like terms: 10 + 1.361111112 = 11.361111112
1.361111112 + 2.333333333x + x2 = 11.361111112

Factor a perfect square on the left side:
(x + 1.166666667)(x + 1.166666667) = 11.361111112

Calculate the square root of the right side: 3.370624736

Break this problem into two subproblems by setting 
(x + 1.166666667) equal to 3.370624736 and -3.370624736.


Subproblem 1
x + 1.166666667 = 3.370624736

Simplifying
x + 1.166666667 = 3.370624736

Reorder the terms:
1.166666667 + x = 3.370624736

Solving
1.166666667 + x = 3.370624736

Solving for variable 'x'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + x = 3.370624736 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + x = 3.370624736 + -1.166666667
x = 3.370624736 + -1.166666667

Combine like terms: 3.370624736 + -1.166666667 = 2.203958069
x = 2.203958069

Simplifying
x = 2.203958069


Subproblem 2
x + 1.166666667 = -3.370624736

Simplifying
x + 1.166666667 = -3.370624736

Reorder the terms:
1.166666667 + x = -3.370624736

Solving
1.166666667 + x = -3.370624736

Solving for variable 'x'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + x = -3.370624736 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + x = -3.370624736 + -1.166666667
x = -3.370624736 + -1.166666667

Combine like terms: -3.370624736 + -1.166666667 = -4.537291403
x = -4.537291403

Simplifying
x = -4.537291403


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.203958069, -4.537291403}

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