7a^2+53a=28

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Solution for 7a^2+53a=28 equation:


Simplifying
7a2 + 53a = 28

Reorder the terms:
53a + 7a2 = 28

Solving
53a + 7a2 = 28

Solving for variable 'a'.

Reorder the terms:
-28 + 53a + 7a2 = 28 + -28

Combine like terms: 28 + -28 = 0
-28 + 53a + 7a2 = 0

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

Divide each side by '7'.
-4 + 7.571428571a + a2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 7.571428571a + 4 + a2 = 0 + 4

Reorder the terms:
-4 + 4 + 7.571428571a + a2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 7.571428571a + a2 = 0 + 4
7.571428571a + a2 = 0 + 4

Combine like terms: 0 + 4 = 4
7.571428571a + a2 = 4

The a term is 7.571428571a.  Take half its coefficient (3.785714286).
Square it (14.33163266) and add it to both sides.

Add '14.33163266' to each side of the equation.
7.571428571a + 14.33163266 + a2 = 4 + 14.33163266

Reorder the terms:
14.33163266 + 7.571428571a + a2 = 4 + 14.33163266

Combine like terms: 4 + 14.33163266 = 18.33163266
14.33163266 + 7.571428571a + a2 = 18.33163266

Factor a perfect square on the left side:
(a + 3.785714286)(a + 3.785714286) = 18.33163266

Calculate the square root of the right side: 4.281545592

Break this problem into two subproblems by setting 
(a + 3.785714286) equal to 4.281545592 and -4.281545592.

Subproblem 1

a + 3.785714286 = 4.281545592 Simplifying a + 3.785714286 = 4.281545592 Reorder the terms: 3.785714286 + a = 4.281545592 Solving 3.785714286 + a = 4.281545592 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.785714286' to each side of the equation. 3.785714286 + -3.785714286 + a = 4.281545592 + -3.785714286 Combine like terms: 3.785714286 + -3.785714286 = 0.000000000 0.000000000 + a = 4.281545592 + -3.785714286 a = 4.281545592 + -3.785714286 Combine like terms: 4.281545592 + -3.785714286 = 0.495831306 a = 0.495831306 Simplifying a = 0.495831306

Subproblem 2

a + 3.785714286 = -4.281545592 Simplifying a + 3.785714286 = -4.281545592 Reorder the terms: 3.785714286 + a = -4.281545592 Solving 3.785714286 + a = -4.281545592 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.785714286' to each side of the equation. 3.785714286 + -3.785714286 + a = -4.281545592 + -3.785714286 Combine like terms: 3.785714286 + -3.785714286 = 0.000000000 0.000000000 + a = -4.281545592 + -3.785714286 a = -4.281545592 + -3.785714286 Combine like terms: -4.281545592 + -3.785714286 = -8.067259878 a = -8.067259878 Simplifying a = -8.067259878

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.495831306, -8.067259878}

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