4x^2+21x+16=0

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Solution for 4x^2+21x+16=0 equation:


Simplifying
4x2 + 21x + 16 = 0

Reorder the terms:
16 + 21x + 4x2 = 0

Solving
16 + 21x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
4 + 5.25x + x2 = 0

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + 5.25x + -4 + x2 = 0 + -4

Reorder the terms:
4 + -4 + 5.25x + x2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + 5.25x + x2 = 0 + -4
5.25x + x2 = 0 + -4

Combine like terms: 0 + -4 = -4
5.25x + x2 = -4

The x term is 5.25x.  Take half its coefficient (2.625).
Square it (6.890625) and add it to both sides.

Add '6.890625' to each side of the equation.
5.25x + 6.890625 + x2 = -4 + 6.890625

Reorder the terms:
6.890625 + 5.25x + x2 = -4 + 6.890625

Combine like terms: -4 + 6.890625 = 2.890625
6.890625 + 5.25x + x2 = 2.890625

Factor a perfect square on the left side:
(x + 2.625)(x + 2.625) = 2.890625

Calculate the square root of the right side: 1.700183814

Break this problem into two subproblems by setting 
(x + 2.625) equal to 1.700183814 and -1.700183814.

Subproblem 1

x + 2.625 = 1.700183814 Simplifying x + 2.625 = 1.700183814 Reorder the terms: 2.625 + x = 1.700183814 Solving 2.625 + x = 1.700183814 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.625' to each side of the equation. 2.625 + -2.625 + x = 1.700183814 + -2.625 Combine like terms: 2.625 + -2.625 = 0.000 0.000 + x = 1.700183814 + -2.625 x = 1.700183814 + -2.625 Combine like terms: 1.700183814 + -2.625 = -0.924816186 x = -0.924816186 Simplifying x = -0.924816186

Subproblem 2

x + 2.625 = -1.700183814 Simplifying x + 2.625 = -1.700183814 Reorder the terms: 2.625 + x = -1.700183814 Solving 2.625 + x = -1.700183814 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.625' to each side of the equation. 2.625 + -2.625 + x = -1.700183814 + -2.625 Combine like terms: 2.625 + -2.625 = 0.000 0.000 + x = -1.700183814 + -2.625 x = -1.700183814 + -2.625 Combine like terms: -1.700183814 + -2.625 = -4.325183814 x = -4.325183814 Simplifying x = -4.325183814

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.924816186, -4.325183814}

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