(a+70)(a+30)=0.5(a)

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Solution for (a+70)(a+30)=0.5(a) equation: 


Simplifying
(a + 70)(a + 30) = 0.5(a)

Reorder the terms:
(70 + a)(a + 30) = 0.5(a)

Reorder the terms:
(70 + a)(30 + a) = 0.5(a)

Multiply (70 + a) * (30 + a)
(70(30 + a) + a(30 + a)) = 0.5(a)
((30 * 70 + a * 70) + a(30 + a)) = 0.5(a)
((2100 + 70a) + a(30 + a)) = 0.5(a)
(2100 + 70a + (30 * a + a * a)) = 0.5(a)
(2100 + 70a + (30a + a2)) = 0.5(a)

Combine like terms: 70a + 30a = 100a
(2100 + 100a + a2) = 0.5(a)

Solving
2100 + 100a + a2 = 0.5a

Solving for variable 'a'.

Reorder the terms:
2100 + 100a + -0.5a + a2 = 0.5a + -0.5a

Combine like terms: 100a + -0.5a = 99.5a
2100 + 99.5a + a2 = 0.5a + -0.5a

Combine like terms: 0.5a + -0.5a = 0.0
2100 + 99.5a + a2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '-2100' to each side of the equation.
2100 + 99.5a + -2100 + a2 = 0.0 + -2100

Reorder the terms:
2100 + -2100 + 99.5a + a2 = 0.0 + -2100

Combine like terms: 2100 + -2100 = 0
0 + 99.5a + a2 = 0.0 + -2100
99.5a + a2 = 0.0 + -2100

Combine like terms: 0.0 + -2100 = -2100
99.5a + a2 = -2100

The a term is 99.5a.  Take half its coefficient (49.75).
Square it (2475.0625) and add it to both sides.

Add '2475.0625' to each side of the equation.
99.5a + 2475.0625 + a2 = -2100 + 2475.0625

Reorder the terms:
2475.0625 + 99.5a + a2 = -2100 + 2475.0625

Combine like terms: -2100 + 2475.0625 = 375.0625
2475.0625 + 99.5a + a2 = 375.0625

Factor a perfect square on the left side:
(a + 49.75)(a + 49.75) = 375.0625

Calculate the square root of the right side: 19.366530407

Break this problem into two subproblems by setting 
(a + 49.75) equal to 19.366530407 and -19.366530407.


Subproblem 1
a + 49.75 = 19.366530407

Simplifying
a + 49.75 = 19.366530407

Reorder the terms:
49.75 + a = 19.366530407

Solving
49.75 + a = 19.366530407

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '-49.75' to each side of the equation.
49.75 + -49.75 + a = 19.366530407 + -49.75

Combine like terms: 49.75 + -49.75 = 0.00
0.00 + a = 19.366530407 + -49.75
a = 19.366530407 + -49.75

Combine like terms: 19.366530407 + -49.75 = -30.383469593
a = -30.383469593

Simplifying
a = -30.383469593


Subproblem 2
a + 49.75 = -19.366530407

Simplifying
a + 49.75 = -19.366530407

Reorder the terms:
49.75 + a = -19.366530407

Solving
49.75 + a = -19.366530407

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '-49.75' to each side of the equation.
49.75 + -49.75 + a = -19.366530407 + -49.75

Combine like terms: 49.75 + -49.75 = 0.00
0.00 + a = -19.366530407 + -49.75
a = -19.366530407 + -49.75

Combine like terms: -19.366530407 + -49.75 = -69.116530407
a = -69.116530407

Simplifying
a = -69.116530407


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-30.383469593, -69.116530407}

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