a^2+15a+45=0

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Solution for a^2+15a+45=0 equation:


Simplifying
a2 + 15a + 45 = 0

Reorder the terms:
45 + 15a + a2 = 0

Solving
45 + 15a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '-45' to each side of the equation.
45 + 15a + -45 + a2 = 0 + -45

Reorder the terms:
45 + -45 + 15a + a2 = 0 + -45

Combine like terms: 45 + -45 = 0
0 + 15a + a2 = 0 + -45
15a + a2 = 0 + -45

Combine like terms: 0 + -45 = -45
15a + a2 = -45

The a term is 15a.  Take half its coefficient (7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
15a + 56.25 + a2 = -45 + 56.25

Reorder the terms:
56.25 + 15a + a2 = -45 + 56.25

Combine like terms: -45 + 56.25 = 11.25
56.25 + 15a + a2 = 11.25

Factor a perfect square on the left side:
(a + 7.5)(a + 7.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(a + 7.5) equal to 3.354101966 and -3.354101966.

Subproblem 1

a + 7.5 = 3.354101966 Simplifying a + 7.5 = 3.354101966 Reorder the terms: 7.5 + a = 3.354101966 Solving 7.5 + a = 3.354101966 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + a = 3.354101966 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + a = 3.354101966 + -7.5 a = 3.354101966 + -7.5 Combine like terms: 3.354101966 + -7.5 = -4.145898034 a = -4.145898034 Simplifying a = -4.145898034

Subproblem 2

a + 7.5 = -3.354101966 Simplifying a + 7.5 = -3.354101966 Reorder the terms: 7.5 + a = -3.354101966 Solving 7.5 + a = -3.354101966 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + a = -3.354101966 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + a = -3.354101966 + -7.5 a = -3.354101966 + -7.5 Combine like terms: -3.354101966 + -7.5 = -10.854101966 a = -10.854101966 Simplifying a = -10.854101966

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-4.145898034, -10.854101966}

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