3a^2+4a-18=0

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Solution for 3a^2+4a-18=0 equation:


Simplifying
3a2 + 4a + -18 = 0

Reorder the terms:
-18 + 4a + 3a2 = 0

Solving
-18 + 4a + 3a2 = 0

Solving for variable 'a'.

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

Divide each side by '3'.
-6 + 1.333333333a + a2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 1.333333333a + 6 + a2 = 0 + 6

Reorder the terms:
-6 + 6 + 1.333333333a + a2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 1.333333333a + a2 = 0 + 6
1.333333333a + a2 = 0 + 6

Combine like terms: 0 + 6 = 6
1.333333333a + a2 = 6

The a term is 1.333333333a.  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.
1.333333333a + 0.4444444442 + a2 = 6 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333a + a2 = 6 + 0.4444444442

Combine like terms: 6 + 0.4444444442 = 6.4444444442
0.4444444442 + 1.333333333a + a2 = 6.4444444442

Factor a perfect square on the left side:
(a + 0.6666666665)(a + 0.6666666665) = 6.4444444442

Calculate the square root of the right side: 2.538591035

Break this problem into two subproblems by setting 
(a + 0.6666666665) equal to 2.538591035 and -2.538591035.

Subproblem 1

a + 0.6666666665 = 2.538591035 Simplifying a + 0.6666666665 = 2.538591035 Reorder the terms: 0.6666666665 + a = 2.538591035 Solving 0.6666666665 + a = 2.538591035 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + a = 2.538591035 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + a = 2.538591035 + -0.6666666665 a = 2.538591035 + -0.6666666665 Combine like terms: 2.538591035 + -0.6666666665 = 1.8719243685 a = 1.8719243685 Simplifying a = 1.8719243685

Subproblem 2

a + 0.6666666665 = -2.538591035 Simplifying a + 0.6666666665 = -2.538591035 Reorder the terms: 0.6666666665 + a = -2.538591035 Solving 0.6666666665 + a = -2.538591035 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + a = -2.538591035 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + a = -2.538591035 + -0.6666666665 a = -2.538591035 + -0.6666666665 Combine like terms: -2.538591035 + -0.6666666665 = -3.2052577015 a = -3.2052577015 Simplifying a = -3.2052577015

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.8719243685, -3.2052577015}

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