30-3x^2-19x=0

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Solution for 30-3x^2-19x=0 equation:


Simplifying
30 + -3x2 + -19x = 0

Reorder the terms:
30 + -19x + -3x2 = 0

Solving
30 + -19x + -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'.
-10 + 6.333333333x + x2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 6.333333333x + 10 + x2 = 0 + 10

Reorder the terms:
-10 + 10 + 6.333333333x + x2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 6.333333333x + x2 = 0 + 10
6.333333333x + x2 = 0 + 10

Combine like terms: 0 + 10 = 10
6.333333333x + x2 = 10

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 = 10 + 10.02777778

Reorder the terms:
10.02777778 + 6.333333333x + x2 = 10 + 10.02777778

Combine like terms: 10 + 10.02777778 = 20.02777778
10.02777778 + 6.333333333x + x2 = 20.02777778

Factor a perfect square on the left side:
(x + 3.166666667)(x + 3.166666667) = 20.02777778

Calculate the square root of the right side: 4.475240528

Break this problem into two subproblems by setting 
(x + 3.166666667) equal to 4.475240528 and -4.475240528.

Subproblem 1

x + 3.166666667 = 4.475240528 Simplifying x + 3.166666667 = 4.475240528 Reorder the terms: 3.166666667 + x = 4.475240528 Solving 3.166666667 + x = 4.475240528 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 = 4.475240528 + -3.166666667 Combine like terms: 3.166666667 + -3.166666667 = 0.000000000 0.000000000 + x = 4.475240528 + -3.166666667 x = 4.475240528 + -3.166666667 Combine like terms: 4.475240528 + -3.166666667 = 1.308573861 x = 1.308573861 Simplifying x = 1.308573861

Subproblem 2

x + 3.166666667 = -4.475240528 Simplifying x + 3.166666667 = -4.475240528 Reorder the terms: 3.166666667 + x = -4.475240528 Solving 3.166666667 + x = -4.475240528 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 = -4.475240528 + -3.166666667 Combine like terms: 3.166666667 + -3.166666667 = 0.000000000 0.000000000 + x = -4.475240528 + -3.166666667 x = -4.475240528 + -3.166666667 Combine like terms: -4.475240528 + -3.166666667 = -7.641907195 x = -7.641907195 Simplifying x = -7.641907195

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.308573861, -7.641907195}

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