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

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

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

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

Reorder the terms:
2 + -7a + a2 = 3 + a

Solving
2 + -7a + a2 = 3 + a

Solving for variable 'a'.

Reorder the terms:
2 + -3 + -7a + -1a + a2 = 3 + a + -3 + -1a

Combine like terms: 2 + -3 = -1
-1 + -7a + -1a + a2 = 3 + a + -3 + -1a

Combine like terms: -7a + -1a = -8a
-1 + -8a + a2 = 3 + a + -3 + -1a

Reorder the terms:
-1 + -8a + a2 = 3 + -3 + a + -1a

Combine like terms: 3 + -3 = 0
-1 + -8a + a2 = 0 + a + -1a
-1 + -8a + a2 = a + -1a

Combine like terms: a + -1a = 0
-1 + -8a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -8a + 1 + a2 = 0 + 1

Reorder the terms:
-1 + 1 + -8a + a2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -8a + a2 = 0 + 1
-8a + a2 = 0 + 1

Combine like terms: 0 + 1 = 1
-8a + a2 = 1

The a term is -8a.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8a + 16 + a2 = 1 + 16

Reorder the terms:
16 + -8a + a2 = 1 + 16

Combine like terms: 1 + 16 = 17
16 + -8a + a2 = 17

Factor a perfect square on the left side:
(a + -4)(a + -4) = 17

Calculate the square root of the right side: 4.123105626

Break this problem into two subproblems by setting 
(a + -4) equal to 4.123105626 and -4.123105626.

Subproblem 1
a + -4 = 4.123105626

Simplifying
a + -4 = 4.123105626

Reorder the terms:
-4 + a = 4.123105626

Solving
-4 + a = 4.123105626

Solving for variable 'a'.

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

Add '4' to each side of the equation.
-4 + 4 + a = 4.123105626 + 4

Combine like terms: -4 + 4 = 0
0 + a = 4.123105626 + 4
a = 4.123105626 + 4

Combine like terms: 4.123105626 + 4 = 8.123105626
a = 8.123105626

Simplifying
a = 8.123105626

Subproblem 2
a + -4 = -4.123105626

Simplifying
a + -4 = -4.123105626

Reorder the terms:
-4 + a = -4.123105626

Solving
-4 + a = -4.123105626

Solving for variable 'a'.

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

Add '4' to each side of the equation.
-4 + 4 + a = -4.123105626 + 4

Combine like terms: -4 + 4 = 0
0 + a = -4.123105626 + 4
a = -4.123105626 + 4

Combine like terms: -4.123105626 + 4 = -0.123105626
a = -0.123105626

Simplifying
a = -0.123105626

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {8.123105626, -0.123105626}