a^2=40-13a

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Solution for a^2=40-13a equation:


Simplifying
a2 = 40 + -13a

Solving
a2 = 40 + -13a

Solving for variable 'a'.

Reorder the terms:
-40 + 13a + a2 = 40 + -13a + -40 + 13a

Reorder the terms:
-40 + 13a + a2 = 40 + -40 + -13a + 13a

Combine like terms: 40 + -40 = 0
-40 + 13a + a2 = 0 + -13a + 13a
-40 + 13a + a2 = -13a + 13a

Combine like terms: -13a + 13a = 0
-40 + 13a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + 13a + 40 + a2 = 0 + 40

Reorder the terms:
-40 + 40 + 13a + a2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + 13a + a2 = 0 + 40
13a + a2 = 0 + 40

Combine like terms: 0 + 40 = 40
13a + a2 = 40

The a term is 13a.  Take half its coefficient (6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
13a + 42.25 + a2 = 40 + 42.25

Reorder the terms:
42.25 + 13a + a2 = 40 + 42.25

Combine like terms: 40 + 42.25 = 82.25
42.25 + 13a + a2 = 82.25

Factor a perfect square on the left side:
(a + 6.5)(a + 6.5) = 82.25

Calculate the square root of the right side: 9.069178574

Break this problem into two subproblems by setting 
(a + 6.5) equal to 9.069178574 and -9.069178574.

Subproblem 1

a + 6.5 = 9.069178574 Simplifying a + 6.5 = 9.069178574 Reorder the terms: 6.5 + a = 9.069178574 Solving 6.5 + a = 9.069178574 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + a = 9.069178574 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + a = 9.069178574 + -6.5 a = 9.069178574 + -6.5 Combine like terms: 9.069178574 + -6.5 = 2.569178574 a = 2.569178574 Simplifying a = 2.569178574

Subproblem 2

a + 6.5 = -9.069178574 Simplifying a + 6.5 = -9.069178574 Reorder the terms: 6.5 + a = -9.069178574 Solving 6.5 + a = -9.069178574 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + a = -9.069178574 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + a = -9.069178574 + -6.5 a = -9.069178574 + -6.5 Combine like terms: -9.069178574 + -6.5 = -15.569178574 a = -15.569178574 Simplifying a = -15.569178574

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.569178574, -15.569178574}

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