a^2+7a=9

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Solution for a^2+7a=9 equation:


Simplifying
a2 + 7a = 9

Reorder the terms:
7a + a2 = 9

Solving
7a + a2 = 9

Solving for variable 'a'.

Reorder the terms:
-9 + 7a + a2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 7a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 7a + 9 + a2 = 0 + 9

Reorder the terms:
-9 + 9 + 7a + a2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 7a + a2 = 0 + 9
7a + a2 = 0 + 9

Combine like terms: 0 + 9 = 9
7a + a2 = 9

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 = 9 + 12.25

Reorder the terms:
12.25 + 7a + a2 = 9 + 12.25

Combine like terms: 9 + 12.25 = 21.25
12.25 + 7a + a2 = 21.25

Factor a perfect square on the left side:
(a + 3.5)(a + 3.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(a + 3.5) equal to 4.609772229 and -4.609772229.

Subproblem 1

a + 3.5 = 4.609772229 Simplifying a + 3.5 = 4.609772229 Reorder the terms: 3.5 + a = 4.609772229 Solving 3.5 + a = 4.609772229 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 = 4.609772229 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + a = 4.609772229 + -3.5 a = 4.609772229 + -3.5 Combine like terms: 4.609772229 + -3.5 = 1.109772229 a = 1.109772229 Simplifying a = 1.109772229

Subproblem 2

a + 3.5 = -4.609772229 Simplifying a + 3.5 = -4.609772229 Reorder the terms: 3.5 + a = -4.609772229 Solving 3.5 + a = -4.609772229 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 = -4.609772229 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + a = -4.609772229 + -3.5 a = -4.609772229 + -3.5 Combine like terms: -4.609772229 + -3.5 = -8.109772229 a = -8.109772229 Simplifying a = -8.109772229

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.109772229, -8.109772229}

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