2a+11=a(a-3)

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Solution for 2a+11=a(a-3) equation:


Simplifying
2a + 11 = a(a + -3)

Reorder the terms:
11 + 2a = a(a + -3)

Reorder the terms:
11 + 2a = a(-3 + a)
11 + 2a = (-3 * a + a * a)
11 + 2a = (-3a + a2)

Solving
11 + 2a = -3a + a2

Solving for variable 'a'.

Combine like terms: 2a + 3a = 5a
11 + 5a + -1a2 = -3a + a2 + 3a + -1a2

Reorder the terms:
11 + 5a + -1a2 = -3a + 3a + a2 + -1a2

Combine like terms: -3a + 3a = 0
11 + 5a + -1a2 = 0 + a2 + -1a2
11 + 5a + -1a2 = a2 + -1a2

Combine like terms: a2 + -1a2 = 0
11 + 5a + -1a2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-11 + -5a + a2 = 0

Move the constant term to the right:

Add '11' to each side of the equation.
-11 + -5a + 11 + a2 = 0 + 11

Reorder the terms:
-11 + 11 + -5a + a2 = 0 + 11

Combine like terms: -11 + 11 = 0
0 + -5a + a2 = 0 + 11
-5a + a2 = 0 + 11

Combine like terms: 0 + 11 = 11
-5a + a2 = 11

The a term is -5a.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5a + 6.25 + a2 = 11 + 6.25

Reorder the terms:
6.25 + -5a + a2 = 11 + 6.25

Combine like terms: 11 + 6.25 = 17.25
6.25 + -5a + a2 = 17.25

Factor a perfect square on the left side:
(a + -2.5)(a + -2.5) = 17.25

Calculate the square root of the right side: 4.153311931

Break this problem into two subproblems by setting 
(a + -2.5) equal to 4.153311931 and -4.153311931.

Subproblem 1

a + -2.5 = 4.153311931 Simplifying a + -2.5 = 4.153311931 Reorder the terms: -2.5 + a = 4.153311931 Solving -2.5 + a = 4.153311931 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + a = 4.153311931 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + a = 4.153311931 + 2.5 a = 4.153311931 + 2.5 Combine like terms: 4.153311931 + 2.5 = 6.653311931 a = 6.653311931 Simplifying a = 6.653311931

Subproblem 2

a + -2.5 = -4.153311931 Simplifying a + -2.5 = -4.153311931 Reorder the terms: -2.5 + a = -4.153311931 Solving -2.5 + a = -4.153311931 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + a = -4.153311931 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + a = -4.153311931 + 2.5 a = -4.153311931 + 2.5 Combine like terms: -4.153311931 + 2.5 = -1.653311931 a = -1.653311931 Simplifying a = -1.653311931

Solution

The solution to the problem is based on the solutions from the subproblems. a = {6.653311931, -1.653311931}

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