7x^2+53x+21=0

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Solution for 7x^2+53x+21=0 equation:


Simplifying
7x2 + 53x + 21 = 0

Reorder the terms:
21 + 53x + 7x2 = 0

Solving
21 + 53x + 7x2 = 0

Solving for variable 'x'.

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

Divide each side by '7'.
3 + 7.571428571x + x2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 7.571428571x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + 7.571428571x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 7.571428571x + x2 = 0 + -3
7.571428571x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
7.571428571x + x2 = -3

The x term is 7.571428571x.  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.
7.571428571x + 14.33163266 + x2 = -3 + 14.33163266

Reorder the terms:
14.33163266 + 7.571428571x + x2 = -3 + 14.33163266

Combine like terms: -3 + 14.33163266 = 11.33163266
14.33163266 + 7.571428571x + x2 = 11.33163266

Factor a perfect square on the left side:
(x + 3.785714286)(x + 3.785714286) = 11.33163266

Calculate the square root of the right side: 3.366249049

Break this problem into two subproblems by setting 
(x + 3.785714286) equal to 3.366249049 and -3.366249049.

Subproblem 1

x + 3.785714286 = 3.366249049 Simplifying x + 3.785714286 = 3.366249049 Reorder the terms: 3.785714286 + x = 3.366249049 Solving 3.785714286 + x = 3.366249049 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.785714286' to each side of the equation. 3.785714286 + -3.785714286 + x = 3.366249049 + -3.785714286 Combine like terms: 3.785714286 + -3.785714286 = 0.000000000 0.000000000 + x = 3.366249049 + -3.785714286 x = 3.366249049 + -3.785714286 Combine like terms: 3.366249049 + -3.785714286 = -0.419465237 x = -0.419465237 Simplifying x = -0.419465237

Subproblem 2

x + 3.785714286 = -3.366249049 Simplifying x + 3.785714286 = -3.366249049 Reorder the terms: 3.785714286 + x = -3.366249049 Solving 3.785714286 + x = -3.366249049 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.785714286' to each side of the equation. 3.785714286 + -3.785714286 + x = -3.366249049 + -3.785714286 Combine like terms: 3.785714286 + -3.785714286 = 0.000000000 0.000000000 + x = -3.366249049 + -3.785714286 x = -3.366249049 + -3.785714286 Combine like terms: -3.366249049 + -3.785714286 = -7.151963335 x = -7.151963335 Simplifying x = -7.151963335

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.419465237, -7.151963335}

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