a^2+9a=2a-9

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


Simplifying
a2 + 9a = 2a + -9

Reorder the terms:
9a + a2 = 2a + -9

Reorder the terms:
9a + a2 = -9 + 2a

Solving
9a + a2 = -9 + 2a

Solving for variable 'a'.

Reorder the terms:
9 + 9a + -2a + a2 = -9 + 2a + 9 + -2a

Combine like terms: 9a + -2a = 7a
9 + 7a + a2 = -9 + 2a + 9 + -2a

Reorder the terms:
9 + 7a + a2 = -9 + 9 + 2a + -2a

Combine like terms: -9 + 9 = 0
9 + 7a + a2 = 0 + 2a + -2a
9 + 7a + a2 = 2a + -2a

Combine like terms: 2a + -2a = 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 = 3.25
12.25 + 7a + a2 = 3.25

Factor a perfect square on the left side:
(a + 3.5)(a + 3.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(a + 3.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

a + 3.5 = 1.802775638 Simplifying a + 3.5 = 1.802775638 Reorder the terms: 3.5 + a = 1.802775638 Solving 3.5 + a = 1.802775638 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 = 1.802775638 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + a = 1.802775638 + -3.5 a = 1.802775638 + -3.5 Combine like terms: 1.802775638 + -3.5 = -1.697224362 a = -1.697224362 Simplifying a = -1.697224362

Subproblem 2

a + 3.5 = -1.802775638 Simplifying a + 3.5 = -1.802775638 Reorder the terms: 3.5 + a = -1.802775638 Solving 3.5 + a = -1.802775638 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 = -1.802775638 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + a = -1.802775638 + -3.5 a = -1.802775638 + -3.5 Combine like terms: -1.802775638 + -3.5 = -5.302775638 a = -5.302775638 Simplifying a = -5.302775638

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-1.697224362, -5.302775638}

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