4x^2+71x+204=0

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Solution for 4x^2+71x+204=0 equation:


Simplifying
4x2 + 71x + 204 = 0

Reorder the terms:
204 + 71x + 4x2 = 0

Solving
204 + 71x + 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'.
51 + 17.75x + x2 = 0

Move the constant term to the right:

Add '-51' to each side of the equation.
51 + 17.75x + -51 + x2 = 0 + -51

Reorder the terms:
51 + -51 + 17.75x + x2 = 0 + -51

Combine like terms: 51 + -51 = 0
0 + 17.75x + x2 = 0 + -51
17.75x + x2 = 0 + -51

Combine like terms: 0 + -51 = -51
17.75x + x2 = -51

The x term is 17.75x.  Take half its coefficient (8.875).
Square it (78.765625) and add it to both sides.

Add '78.765625' to each side of the equation.
17.75x + 78.765625 + x2 = -51 + 78.765625

Reorder the terms:
78.765625 + 17.75x + x2 = -51 + 78.765625

Combine like terms: -51 + 78.765625 = 27.765625
78.765625 + 17.75x + x2 = 27.765625

Factor a perfect square on the left side:
(x + 8.875)(x + 8.875) = 27.765625

Calculate the square root of the right side: 5.269309727

Break this problem into two subproblems by setting 
(x + 8.875) equal to 5.269309727 and -5.269309727.

Subproblem 1

x + 8.875 = 5.269309727 Simplifying x + 8.875 = 5.269309727 Reorder the terms: 8.875 + x = 5.269309727 Solving 8.875 + x = 5.269309727 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.875' to each side of the equation. 8.875 + -8.875 + x = 5.269309727 + -8.875 Combine like terms: 8.875 + -8.875 = 0.000 0.000 + x = 5.269309727 + -8.875 x = 5.269309727 + -8.875 Combine like terms: 5.269309727 + -8.875 = -3.605690273 x = -3.605690273 Simplifying x = -3.605690273

Subproblem 2

x + 8.875 = -5.269309727 Simplifying x + 8.875 = -5.269309727 Reorder the terms: 8.875 + x = -5.269309727 Solving 8.875 + x = -5.269309727 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.875' to each side of the equation. 8.875 + -8.875 + x = -5.269309727 + -8.875 Combine like terms: 8.875 + -8.875 = 0.000 0.000 + x = -5.269309727 + -8.875 x = -5.269309727 + -8.875 Combine like terms: -5.269309727 + -8.875 = -14.144309727 x = -14.144309727 Simplifying x = -14.144309727

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.605690273, -14.144309727}

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