3x^2+21x+16=0

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Solution for 3x^2+21x+16=0 equation:


Simplifying
3x2 + 21x + 16 = 0

Reorder the terms:
16 + 21x + 3x2 = 0

Solving
16 + 21x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
5.333333333 + 7x + x2 = 0

Move the constant term to the right:

Add '-5.333333333' to each side of the equation.
5.333333333 + 7x + -5.333333333 + x2 = 0 + -5.333333333

Reorder the terms:
5.333333333 + -5.333333333 + 7x + x2 = 0 + -5.333333333

Combine like terms: 5.333333333 + -5.333333333 = 0.000000000
0.000000000 + 7x + x2 = 0 + -5.333333333
7x + x2 = 0 + -5.333333333

Combine like terms: 0 + -5.333333333 = -5.333333333
7x + x2 = -5.333333333

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = -5.333333333 + 12.25

Reorder the terms:
12.25 + 7x + x2 = -5.333333333 + 12.25

Combine like terms: -5.333333333 + 12.25 = 6.916666667
12.25 + 7x + x2 = 6.916666667

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 6.916666667

Calculate the square root of the right side: 2.62995564

Break this problem into two subproblems by setting 
(x + 3.5) equal to 2.62995564 and -2.62995564.

Subproblem 1

x + 3.5 = 2.62995564 Simplifying x + 3.5 = 2.62995564 Reorder the terms: 3.5 + x = 2.62995564 Solving 3.5 + x = 2.62995564 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 2.62995564 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 2.62995564 + -3.5 x = 2.62995564 + -3.5 Combine like terms: 2.62995564 + -3.5 = -0.87004436 x = -0.87004436 Simplifying x = -0.87004436

Subproblem 2

x + 3.5 = -2.62995564 Simplifying x + 3.5 = -2.62995564 Reorder the terms: 3.5 + x = -2.62995564 Solving 3.5 + x = -2.62995564 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -2.62995564 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -2.62995564 + -3.5 x = -2.62995564 + -3.5 Combine like terms: -2.62995564 + -3.5 = -6.12995564 x = -6.12995564 Simplifying x = -6.12995564

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.87004436, -6.12995564}

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