2a+3a(a-7)=a+3

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Solution for 2a+3a(a-7)=a+3 equation:


Simplifying
2a + 3a(a + -7) = a + 3

Reorder the terms:
2a + 3a(-7 + a) = a + 3
2a + (-7 * 3a + a * 3a) = a + 3
2a + (-21a + 3a2) = a + 3

Combine like terms: 2a + -21a = -19a
-19a + 3a2 = a + 3

Reorder the terms:
-19a + 3a2 = 3 + a

Solving
-19a + 3a2 = 3 + a

Solving for variable 'a'.

Reorder the terms:
-3 + -19a + -1a + 3a2 = 3 + a + -3 + -1a

Combine like terms: -19a + -1a = -20a
-3 + -20a + 3a2 = 3 + a + -3 + -1a

Reorder the terms:
-3 + -20a + 3a2 = 3 + -3 + a + -1a

Combine like terms: 3 + -3 = 0
-3 + -20a + 3a2 = 0 + a + -1a
-3 + -20a + 3a2 = a + -1a

Combine like terms: a + -1a = 0
-3 + -20a + 3a2 = 0

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

Divide each side by '3'.
-1 + -6.666666667a + a2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -6.666666667a + 1 + a2 = 0 + 1

Reorder the terms:
-1 + 1 + -6.666666667a + a2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -6.666666667a + a2 = 0 + 1
-6.666666667a + a2 = 0 + 1

Combine like terms: 0 + 1 = 1
-6.666666667a + a2 = 1

The a term is -6.666666667a.  Take half its coefficient (-3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
-6.666666667a + 11.11111112 + a2 = 1 + 11.11111112

Reorder the terms:
11.11111112 + -6.666666667a + a2 = 1 + 11.11111112

Combine like terms: 1 + 11.11111112 = 12.11111112
11.11111112 + -6.666666667a + a2 = 12.11111112

Factor a perfect square on the left side:
(a + -3.333333334)(a + -3.333333334) = 12.11111112

Calculate the square root of the right side: 3.480102171

Break this problem into two subproblems by setting 
(a + -3.333333334) equal to 3.480102171 and -3.480102171.

Subproblem 1

a + -3.333333334 = 3.480102171 Simplifying a + -3.333333334 = 3.480102171 Reorder the terms: -3.333333334 + a = 3.480102171 Solving -3.333333334 + a = 3.480102171 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.333333334' to each side of the equation. -3.333333334 + 3.333333334 + a = 3.480102171 + 3.333333334 Combine like terms: -3.333333334 + 3.333333334 = 0.000000000 0.000000000 + a = 3.480102171 + 3.333333334 a = 3.480102171 + 3.333333334 Combine like terms: 3.480102171 + 3.333333334 = 6.813435505 a = 6.813435505 Simplifying a = 6.813435505

Subproblem 2

a + -3.333333334 = -3.480102171 Simplifying a + -3.333333334 = -3.480102171 Reorder the terms: -3.333333334 + a = -3.480102171 Solving -3.333333334 + a = -3.480102171 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.333333334' to each side of the equation. -3.333333334 + 3.333333334 + a = -3.480102171 + 3.333333334 Combine like terms: -3.333333334 + 3.333333334 = 0.000000000 0.000000000 + a = -3.480102171 + 3.333333334 a = -3.480102171 + 3.333333334 Combine like terms: -3.480102171 + 3.333333334 = -0.146768837 a = -0.146768837 Simplifying a = -0.146768837

Solution

The solution to the problem is based on the solutions from the subproblems. a = {6.813435505, -0.146768837}

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