4a-4(14-a)a=7

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Solution for 4a-4(14-a)a=7 equation:


Simplifying
4a + -4(14 + -1a) * a = 7

Reorder the terms for easier multiplication:
4a + -4a(14 + -1a) = 7
4a + (14 * -4a + -1a * -4a) = 7
4a + (-56a + 4a2) = 7

Combine like terms: 4a + -56a = -52a
-52a + 4a2 = 7

Solving
-52a + 4a2 = 7

Solving for variable 'a'.

Reorder the terms:
-7 + -52a + 4a2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + -52a + 4a2 = 0

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
-1.75 + -13a + a2 = 0

Move the constant term to the right:

Add '1.75' to each side of the equation.
-1.75 + -13a + 1.75 + a2 = 0 + 1.75

Reorder the terms:
-1.75 + 1.75 + -13a + a2 = 0 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + -13a + a2 = 0 + 1.75
-13a + a2 = 0 + 1.75

Combine like terms: 0 + 1.75 = 1.75
-13a + a2 = 1.75

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 = 1.75 + 42.25

Reorder the terms:
42.25 + -13a + a2 = 1.75 + 42.25

Combine like terms: 1.75 + 42.25 = 44
42.25 + -13a + a2 = 44

Factor a perfect square on the left side:
(a + -6.5)(a + -6.5) = 44

Calculate the square root of the right side: 6.633249581

Break this problem into two subproblems by setting 
(a + -6.5) equal to 6.633249581 and -6.633249581.

Subproblem 1

a + -6.5 = 6.633249581 Simplifying a + -6.5 = 6.633249581 Reorder the terms: -6.5 + a = 6.633249581 Solving -6.5 + a = 6.633249581 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 = 6.633249581 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + a = 6.633249581 + 6.5 a = 6.633249581 + 6.5 Combine like terms: 6.633249581 + 6.5 = 13.133249581 a = 13.133249581 Simplifying a = 13.133249581

Subproblem 2

a + -6.5 = -6.633249581 Simplifying a + -6.5 = -6.633249581 Reorder the terms: -6.5 + a = -6.633249581 Solving -6.5 + a = -6.633249581 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 = -6.633249581 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + a = -6.633249581 + 6.5 a = -6.633249581 + 6.5 Combine like terms: -6.633249581 + 6.5 = -0.133249581 a = -0.133249581 Simplifying a = -0.133249581

Solution

The solution to the problem is based on the solutions from the subproblems. a = {13.133249581, -0.133249581}

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