a^2+9a=14

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Solution for a^2+9a=14 equation:


Simplifying
a2 + 9a = 14

Reorder the terms:
9a + a2 = 14

Solving
9a + a2 = 14

Solving for variable 'a'.

Reorder the terms:
-14 + 9a + a2 = 14 + -14

Combine like terms: 14 + -14 = 0
-14 + 9a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + 9a + 14 + a2 = 0 + 14

Reorder the terms:
-14 + 14 + 9a + a2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + 9a + a2 = 0 + 14
9a + a2 = 0 + 14

Combine like terms: 0 + 14 = 14
9a + a2 = 14

The a term is 9a.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9a + 20.25 + a2 = 14 + 20.25

Reorder the terms:
20.25 + 9a + a2 = 14 + 20.25

Combine like terms: 14 + 20.25 = 34.25
20.25 + 9a + a2 = 34.25

Factor a perfect square on the left side:
(a + 4.5)(a + 4.5) = 34.25

Calculate the square root of the right side: 5.852349955

Break this problem into two subproblems by setting 
(a + 4.5) equal to 5.852349955 and -5.852349955.

Subproblem 1

a + 4.5 = 5.852349955 Simplifying a + 4.5 = 5.852349955 Reorder the terms: 4.5 + a = 5.852349955 Solving 4.5 + a = 5.852349955 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + a = 5.852349955 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + a = 5.852349955 + -4.5 a = 5.852349955 + -4.5 Combine like terms: 5.852349955 + -4.5 = 1.352349955 a = 1.352349955 Simplifying a = 1.352349955

Subproblem 2

a + 4.5 = -5.852349955 Simplifying a + 4.5 = -5.852349955 Reorder the terms: 4.5 + a = -5.852349955 Solving 4.5 + a = -5.852349955 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + a = -5.852349955 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + a = -5.852349955 + -4.5 a = -5.852349955 + -4.5 Combine like terms: -5.852349955 + -4.5 = -10.352349955 a = -10.352349955 Simplifying a = -10.352349955

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.352349955, -10.352349955}

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