x^2+21x+7=0

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


Simplifying
x2 + 21x + 7 = 0

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

Solving
7 + 21x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + 21x + -7 + x2 = 0 + -7

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

Combine like terms: 7 + -7 = 0
0 + 21x + x2 = 0 + -7
21x + x2 = 0 + -7

Combine like terms: 0 + -7 = -7
21x + x2 = -7

The x term is 21x.  Take half its coefficient (10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
21x + 110.25 + x2 = -7 + 110.25

Reorder the terms:
110.25 + 21x + x2 = -7 + 110.25

Combine like terms: -7 + 110.25 = 103.25
110.25 + 21x + x2 = 103.25

Factor a perfect square on the left side:
(x + 10.5)(x + 10.5) = 103.25

Calculate the square root of the right side: 10.161200716

Break this problem into two subproblems by setting 
(x + 10.5) equal to 10.161200716 and -10.161200716.

Subproblem 1

x + 10.5 = 10.161200716 Simplifying x + 10.5 = 10.161200716 Reorder the terms: 10.5 + x = 10.161200716 Solving 10.5 + x = 10.161200716 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10.5' to each side of the equation. 10.5 + -10.5 + x = 10.161200716 + -10.5 Combine like terms: 10.5 + -10.5 = 0.0 0.0 + x = 10.161200716 + -10.5 x = 10.161200716 + -10.5 Combine like terms: 10.161200716 + -10.5 = -0.338799284 x = -0.338799284 Simplifying x = -0.338799284

Subproblem 2

x + 10.5 = -10.161200716 Simplifying x + 10.5 = -10.161200716 Reorder the terms: 10.5 + x = -10.161200716 Solving 10.5 + x = -10.161200716 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10.5' to each side of the equation. 10.5 + -10.5 + x = -10.161200716 + -10.5 Combine like terms: 10.5 + -10.5 = 0.0 0.0 + x = -10.161200716 + -10.5 x = -10.161200716 + -10.5 Combine like terms: -10.161200716 + -10.5 = -20.661200716 x = -20.661200716 Simplifying x = -20.661200716

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.338799284, -20.661200716}

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