13a+15=a(a+6)

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Solution for 13a+15=a(a+6) equation:


Simplifying
13a + 15 = a(a + 6)

Reorder the terms:
15 + 13a = a(a + 6)

Reorder the terms:
15 + 13a = a(6 + a)
15 + 13a = (6 * a + a * a)
15 + 13a = (6a + a2)

Solving
15 + 13a = 6a + a2

Solving for variable 'a'.

Combine like terms: 13a + -6a = 7a
15 + 7a + -1a2 = 6a + a2 + -6a + -1a2

Reorder the terms:
15 + 7a + -1a2 = 6a + -6a + a2 + -1a2

Combine like terms: 6a + -6a = 0
15 + 7a + -1a2 = 0 + a2 + -1a2
15 + 7a + -1a2 = a2 + -1a2

Combine like terms: a2 + -1a2 = 0
15 + 7a + -1a2 = 0

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

Divide each side by '-1'.
-15 + -7a + a2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + -7a + 15 + a2 = 0 + 15

Reorder the terms:
-15 + 15 + -7a + a2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + -7a + a2 = 0 + 15
-7a + a2 = 0 + 15

Combine like terms: 0 + 15 = 15
-7a + a2 = 15

The a term is -7a.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7a + 12.25 + a2 = 15 + 12.25

Reorder the terms:
12.25 + -7a + a2 = 15 + 12.25

Combine like terms: 15 + 12.25 = 27.25
12.25 + -7a + a2 = 27.25

Factor a perfect square on the left side:
(a + -3.5)(a + -3.5) = 27.25

Calculate the square root of the right side: 5.220153254

Break this problem into two subproblems by setting 
(a + -3.5) equal to 5.220153254 and -5.220153254.

Subproblem 1

a + -3.5 = 5.220153254 Simplifying a + -3.5 = 5.220153254 Reorder the terms: -3.5 + a = 5.220153254 Solving -3.5 + a = 5.220153254 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + a = 5.220153254 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + a = 5.220153254 + 3.5 a = 5.220153254 + 3.5 Combine like terms: 5.220153254 + 3.5 = 8.720153254 a = 8.720153254 Simplifying a = 8.720153254

Subproblem 2

a + -3.5 = -5.220153254 Simplifying a + -3.5 = -5.220153254 Reorder the terms: -3.5 + a = -5.220153254 Solving -3.5 + a = -5.220153254 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + a = -5.220153254 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + a = -5.220153254 + 3.5 a = -5.220153254 + 3.5 Combine like terms: -5.220153254 + 3.5 = -1.720153254 a = -1.720153254 Simplifying a = -1.720153254

Solution

The solution to the problem is based on the solutions from the subproblems. a = {8.720153254, -1.720153254}

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