7a^2+25a+21=0

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Solution for 7a^2+25a+21=0 equation:


Simplifying
7a2 + 25a + 21 = 0

Reorder the terms:
21 + 25a + 7a2 = 0

Solving
21 + 25a + 7a2 = 0

Solving for variable 'a'.

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

Divide each side by '7'.
3 + 3.571428571a + a2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 3.571428571a + -3 + a2 = 0 + -3

Reorder the terms:
3 + -3 + 3.571428571a + a2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 3.571428571a + a2 = 0 + -3
3.571428571a + a2 = 0 + -3

Combine like terms: 0 + -3 = -3
3.571428571a + a2 = -3

The a term is 3.571428571a.  Take half its coefficient (1.785714286).
Square it (3.188775511) and add it to both sides.

Add '3.188775511' to each side of the equation.
3.571428571a + 3.188775511 + a2 = -3 + 3.188775511

Reorder the terms:
3.188775511 + 3.571428571a + a2 = -3 + 3.188775511

Combine like terms: -3 + 3.188775511 = 0.188775511
3.188775511 + 3.571428571a + a2 = 0.188775511

Factor a perfect square on the left side:
(a + 1.785714286)(a + 1.785714286) = 0.188775511

Calculate the square root of the right side: 0.434483039

Break this problem into two subproblems by setting 
(a + 1.785714286) equal to 0.434483039 and -0.434483039.

Subproblem 1

a + 1.785714286 = 0.434483039 Simplifying a + 1.785714286 = 0.434483039 Reorder the terms: 1.785714286 + a = 0.434483039 Solving 1.785714286 + a = 0.434483039 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.785714286' to each side of the equation. 1.785714286 + -1.785714286 + a = 0.434483039 + -1.785714286 Combine like terms: 1.785714286 + -1.785714286 = 0.000000000 0.000000000 + a = 0.434483039 + -1.785714286 a = 0.434483039 + -1.785714286 Combine like terms: 0.434483039 + -1.785714286 = -1.351231247 a = -1.351231247 Simplifying a = -1.351231247

Subproblem 2

a + 1.785714286 = -0.434483039 Simplifying a + 1.785714286 = -0.434483039 Reorder the terms: 1.785714286 + a = -0.434483039 Solving 1.785714286 + a = -0.434483039 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.785714286' to each side of the equation. 1.785714286 + -1.785714286 + a = -0.434483039 + -1.785714286 Combine like terms: 1.785714286 + -1.785714286 = 0.000000000 0.000000000 + a = -0.434483039 + -1.785714286 a = -0.434483039 + -1.785714286 Combine like terms: -0.434483039 + -1.785714286 = -2.220197325 a = -2.220197325 Simplifying a = -2.220197325

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-1.351231247, -2.220197325}

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