5a(7a+7)=1

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Solution for 5a(7a+7)=1 equation:


Simplifying
5a(7a + 7) = 1

Reorder the terms:
5a(7 + 7a) = 1
(7 * 5a + 7a * 5a) = 1
(35a + 35a2) = 1

Solving
35a + 35a2 = 1

Solving for variable 'a'.

Reorder the terms:
-1 + 35a + 35a2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 35a + 35a2 = 0

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

Divide each side by '35'.
-0.02857142857 + a + a2 = 0

Move the constant term to the right:

Add '0.02857142857' to each side of the equation.
-0.02857142857 + a + 0.02857142857 + a2 = 0 + 0.02857142857

Reorder the terms:
-0.02857142857 + 0.02857142857 + a + a2 = 0 + 0.02857142857

Combine like terms: -0.02857142857 + 0.02857142857 = 0.00000000000
0.00000000000 + a + a2 = 0 + 0.02857142857
a + a2 = 0 + 0.02857142857

Combine like terms: 0 + 0.02857142857 = 0.02857142857
a + a2 = 0.02857142857

The a term is a.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + a2 = 0.02857142857 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + a2 = 0.02857142857 + 0.25

Combine like terms: 0.02857142857 + 0.25 = 0.27857142857
1.25 + a2 = 0.27857142857

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 0.27857142857

Calculate the square root of the right side: 0.527798663

Break this problem into two subproblems by setting 
(a + 0.5) equal to 0.527798663 and -0.527798663.

Subproblem 1

a + 0.5 = 0.527798663 Simplifying a + 0.5 = 0.527798663 Reorder the terms: 0.5 + a = 0.527798663 Solving 0.5 + a = 0.527798663 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 0.527798663 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 0.527798663 + -0.5 a = 0.527798663 + -0.5 Combine like terms: 0.527798663 + -0.5 = 0.027798663 a = 0.027798663 Simplifying a = 0.027798663

Subproblem 2

a + 0.5 = -0.527798663 Simplifying a + 0.5 = -0.527798663 Reorder the terms: 0.5 + a = -0.527798663 Solving 0.5 + a = -0.527798663 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -0.527798663 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -0.527798663 + -0.5 a = -0.527798663 + -0.5 Combine like terms: -0.527798663 + -0.5 = -1.027798663 a = -1.027798663 Simplifying a = -1.027798663

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.027798663, -1.027798663}

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