3x^2+4x=24

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Solution for 3x^2+4x=24 equation:


Simplifying
3x2 + 4x = 24

Reorder the terms:
4x + 3x2 = 24

Solving
4x + 3x2 = 24

Solving for variable 'x'.

Reorder the terms:
-24 + 4x + 3x2 = 24 + -24

Combine like terms: 24 + -24 = 0
-24 + 4x + 3x2 = 0

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

Divide each side by '3'.
-8 + 1.333333333x + x2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 1.333333333x + 8 + x2 = 0 + 8

Reorder the terms:
-8 + 8 + 1.333333333x + x2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 1.333333333x + x2 = 0 + 8
1.333333333x + x2 = 0 + 8

Combine like terms: 0 + 8 = 8
1.333333333x + x2 = 8

The x term is 1.333333333x.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333x + 0.4444444442 + x2 = 8 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333x + x2 = 8 + 0.4444444442

Combine like terms: 8 + 0.4444444442 = 8.4444444442
0.4444444442 + 1.333333333x + x2 = 8.4444444442

Factor a perfect square on the left side:
(x + 0.6666666665)(x + 0.6666666665) = 8.4444444442

Calculate the square root of the right side: 2.905932629

Break this problem into two subproblems by setting 
(x + 0.6666666665) equal to 2.905932629 and -2.905932629.

Subproblem 1

x + 0.6666666665 = 2.905932629 Simplifying x + 0.6666666665 = 2.905932629 Reorder the terms: 0.6666666665 + x = 2.905932629 Solving 0.6666666665 + x = 2.905932629 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = 2.905932629 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = 2.905932629 + -0.6666666665 x = 2.905932629 + -0.6666666665 Combine like terms: 2.905932629 + -0.6666666665 = 2.2392659625 x = 2.2392659625 Simplifying x = 2.2392659625

Subproblem 2

x + 0.6666666665 = -2.905932629 Simplifying x + 0.6666666665 = -2.905932629 Reorder the terms: 0.6666666665 + x = -2.905932629 Solving 0.6666666665 + x = -2.905932629 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = -2.905932629 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = -2.905932629 + -0.6666666665 x = -2.905932629 + -0.6666666665 Combine like terms: -2.905932629 + -0.6666666665 = -3.5725992955 x = -3.5725992955 Simplifying x = -3.5725992955

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.2392659625, -3.5725992955}

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