3x^2+19x=54

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Solution for 3x^2+19x=54 equation:


Simplifying
3x2 + 19x = 54

Reorder the terms:
19x + 3x2 = 54

Solving
19x + 3x2 = 54

Solving for variable 'x'.

Reorder the terms:
-54 + 19x + 3x2 = 54 + -54

Combine like terms: 54 + -54 = 0
-54 + 19x + 3x2 = 0

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

Divide each side by '3'.
-18 + 6.333333333x + x2 = 0

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + 6.333333333x + 18 + x2 = 0 + 18

Reorder the terms:
-18 + 18 + 6.333333333x + x2 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + 6.333333333x + x2 = 0 + 18
6.333333333x + x2 = 0 + 18

Combine like terms: 0 + 18 = 18
6.333333333x + x2 = 18

The x term is 6.333333333x.  Take half its coefficient (3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
6.333333333x + 10.02777778 + x2 = 18 + 10.02777778

Reorder the terms:
10.02777778 + 6.333333333x + x2 = 18 + 10.02777778

Combine like terms: 18 + 10.02777778 = 28.02777778
10.02777778 + 6.333333333x + x2 = 28.02777778

Factor a perfect square on the left side:
(x + 3.166666667)(x + 3.166666667) = 28.02777778

Calculate the square root of the right side: 5.294126725

Break this problem into two subproblems by setting 
(x + 3.166666667) equal to 5.294126725 and -5.294126725.

Subproblem 1

x + 3.166666667 = 5.294126725 Simplifying x + 3.166666667 = 5.294126725 Reorder the terms: 3.166666667 + x = 5.294126725 Solving 3.166666667 + x = 5.294126725 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.166666667' to each side of the equation. 3.166666667 + -3.166666667 + x = 5.294126725 + -3.166666667 Combine like terms: 3.166666667 + -3.166666667 = 0.000000000 0.000000000 + x = 5.294126725 + -3.166666667 x = 5.294126725 + -3.166666667 Combine like terms: 5.294126725 + -3.166666667 = 2.127460058 x = 2.127460058 Simplifying x = 2.127460058

Subproblem 2

x + 3.166666667 = -5.294126725 Simplifying x + 3.166666667 = -5.294126725 Reorder the terms: 3.166666667 + x = -5.294126725 Solving 3.166666667 + x = -5.294126725 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.166666667' to each side of the equation. 3.166666667 + -3.166666667 + x = -5.294126725 + -3.166666667 Combine like terms: 3.166666667 + -3.166666667 = 0.000000000 0.000000000 + x = -5.294126725 + -3.166666667 x = -5.294126725 + -3.166666667 Combine like terms: -5.294126725 + -3.166666667 = -8.460793392 x = -8.460793392 Simplifying x = -8.460793392

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.127460058, -8.460793392}

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