x^2+21x+31=

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Solution for x^2+21x+31= equation:


Simplifying
x2 + 21x + 31 = 0

Reorder the terms:
31 + 21x + x2 = 0

Solving
31 + 21x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-31' to each side of the equation.
31 + 21x + -31 + x2 = 0 + -31

Reorder the terms:
31 + -31 + 21x + x2 = 0 + -31

Combine like terms: 31 + -31 = 0
0 + 21x + x2 = 0 + -31
21x + x2 = 0 + -31

Combine like terms: 0 + -31 = -31
21x + x2 = -31

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 = -31 + 110.25

Reorder the terms:
110.25 + 21x + x2 = -31 + 110.25

Combine like terms: -31 + 110.25 = 79.25
110.25 + 21x + x2 = 79.25

Factor a perfect square on the left side:
(x + 10.5)(x + 10.5) = 79.25

Calculate the square root of the right side: 8.902246907

Break this problem into two subproblems by setting 
(x + 10.5) equal to 8.902246907 and -8.902246907.

Subproblem 1

x + 10.5 = 8.902246907 Simplifying x + 10.5 = 8.902246907 Reorder the terms: 10.5 + x = 8.902246907 Solving 10.5 + x = 8.902246907 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 = 8.902246907 + -10.5 Combine like terms: 10.5 + -10.5 = 0.0 0.0 + x = 8.902246907 + -10.5 x = 8.902246907 + -10.5 Combine like terms: 8.902246907 + -10.5 = -1.597753093 x = -1.597753093 Simplifying x = -1.597753093

Subproblem 2

x + 10.5 = -8.902246907 Simplifying x + 10.5 = -8.902246907 Reorder the terms: 10.5 + x = -8.902246907 Solving 10.5 + x = -8.902246907 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 = -8.902246907 + -10.5 Combine like terms: 10.5 + -10.5 = 0.0 0.0 + x = -8.902246907 + -10.5 x = -8.902246907 + -10.5 Combine like terms: -8.902246907 + -10.5 = -19.402246907 x = -19.402246907 Simplifying x = -19.402246907

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.597753093, -19.402246907}

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