21a^2+5a-7=0

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Solution for 21a^2+5a-7=0 equation: 


Simplifying
21a2 + 5a + -7 = 0

Reorder the terms:
-7 + 5a + 21a2 = 0

Solving
-7 + 5a + 21a2 = 0

Solving for variable 'a'.

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

Divide each side by '21'.
-0.3333333333 + 0.2380952381a + a2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + 0.2380952381a + 0.3333333333 + a2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + 0.2380952381a + a2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + 0.2380952381a + a2 = 0 + 0.3333333333
0.2380952381a + a2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
0.2380952381a + a2 = 0.3333333333

The a term is 0.2380952381a.  Take half its coefficient (0.1190476191).
Square it (0.01417233561) and add it to both sides.

Add '0.01417233561' to each side of the equation.
0.2380952381a + 0.01417233561 + a2 = 0.3333333333 + 0.01417233561

Reorder the terms:
0.01417233561 + 0.2380952381a + a2 = 0.3333333333 + 0.01417233561

Combine like terms: 0.3333333333 + 0.01417233561 = 0.34750566891
0.01417233561 + 0.2380952381a + a2 = 0.34750566891

Factor a perfect square on the left side:
(a + 0.1190476191)(a + 0.1190476191) = 0.34750566891

Calculate the square root of the right side: 0.589496114

Break this problem into two subproblems by setting 
(a + 0.1190476191) equal to 0.589496114 and -0.589496114.


Subproblem 1
a + 0.1190476191 = 0.589496114

Simplifying
a + 0.1190476191 = 0.589496114

Reorder the terms:
0.1190476191 + a = 0.589496114

Solving
0.1190476191 + a = 0.589496114

Solving for variable 'a'.

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

Add '-0.1190476191' to each side of the equation.
0.1190476191 + -0.1190476191 + a = 0.589496114 + -0.1190476191

Combine like terms: 0.1190476191 + -0.1190476191 = 0.0000000000
0.0000000000 + a = 0.589496114 + -0.1190476191
a = 0.589496114 + -0.1190476191

Combine like terms: 0.589496114 + -0.1190476191 = 0.4704484949
a = 0.4704484949

Simplifying
a = 0.4704484949


Subproblem 2
a + 0.1190476191 = -0.589496114

Simplifying
a + 0.1190476191 = -0.589496114

Reorder the terms:
0.1190476191 + a = -0.589496114

Solving
0.1190476191 + a = -0.589496114

Solving for variable 'a'.

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

Add '-0.1190476191' to each side of the equation.
0.1190476191 + -0.1190476191 + a = -0.589496114 + -0.1190476191

Combine like terms: 0.1190476191 + -0.1190476191 = 0.0000000000
0.0000000000 + a = -0.589496114 + -0.1190476191
a = -0.589496114 + -0.1190476191

Combine like terms: -0.589496114 + -0.1190476191 = -0.7085437331
a = -0.7085437331

Simplifying
a = -0.7085437331


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.4704484949, -0.7085437331}

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