4x^2+19x+16=0

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Solution for 4x^2+19x+16=0 equation:


Simplifying
4x2 + 19x + 16 = 0

Reorder the terms:
16 + 19x + 4x2 = 0

Solving
16 + 19x + 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 + 4.75x + x2 = 0

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + 4.75x + -4 + x2 = 0 + -4

Reorder the terms:
4 + -4 + 4.75x + x2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + 4.75x + x2 = 0 + -4
4.75x + x2 = 0 + -4

Combine like terms: 0 + -4 = -4
4.75x + x2 = -4

The x term is 4.75x.  Take half its coefficient (2.375).
Square it (5.640625) and add it to both sides.

Add '5.640625' to each side of the equation.
4.75x + 5.640625 + x2 = -4 + 5.640625

Reorder the terms:
5.640625 + 4.75x + x2 = -4 + 5.640625

Combine like terms: -4 + 5.640625 = 1.640625
5.640625 + 4.75x + x2 = 1.640625

Factor a perfect square on the left side:
(x + 2.375)(x + 2.375) = 1.640625

Calculate the square root of the right side: 1.280868846

Break this problem into two subproblems by setting 
(x + 2.375) equal to 1.280868846 and -1.280868846.

Subproblem 1

x + 2.375 = 1.280868846 Simplifying x + 2.375 = 1.280868846 Reorder the terms: 2.375 + x = 1.280868846 Solving 2.375 + x = 1.280868846 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + x = 1.280868846 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + x = 1.280868846 + -2.375 x = 1.280868846 + -2.375 Combine like terms: 1.280868846 + -2.375 = -1.094131154 x = -1.094131154 Simplifying x = -1.094131154

Subproblem 2

x + 2.375 = -1.280868846 Simplifying x + 2.375 = -1.280868846 Reorder the terms: 2.375 + x = -1.280868846 Solving 2.375 + x = -1.280868846 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + x = -1.280868846 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + x = -1.280868846 + -2.375 x = -1.280868846 + -2.375 Combine like terms: -1.280868846 + -2.375 = -3.655868846 x = -3.655868846 Simplifying x = -3.655868846

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.094131154, -3.655868846}

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