19x^2+12x=32

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Solution for 19x^2+12x=32 equation:


Simplifying
19x2 + 12x = 32

Reorder the terms:
12x + 19x2 = 32

Solving
12x + 19x2 = 32

Solving for variable 'x'.

Reorder the terms:
-32 + 12x + 19x2 = 32 + -32

Combine like terms: 32 + -32 = 0
-32 + 12x + 19x2 = 0

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

Divide each side by '19'.
-1.684210526 + 0.6315789474x + x2 = 0

Move the constant term to the right:

Add '1.684210526' to each side of the equation.
-1.684210526 + 0.6315789474x + 1.684210526 + x2 = 0 + 1.684210526

Reorder the terms:
-1.684210526 + 1.684210526 + 0.6315789474x + x2 = 0 + 1.684210526

Combine like terms: -1.684210526 + 1.684210526 = 0.000000000
0.000000000 + 0.6315789474x + x2 = 0 + 1.684210526
0.6315789474x + x2 = 0 + 1.684210526

Combine like terms: 0 + 1.684210526 = 1.684210526
0.6315789474x + x2 = 1.684210526

The x term is 0.6315789474x.  Take half its coefficient (0.3157894737).
Square it (0.09972299170) and add it to both sides.

Add '0.09972299170' to each side of the equation.
0.6315789474x + 0.09972299170 + x2 = 1.684210526 + 0.09972299170

Reorder the terms:
0.09972299170 + 0.6315789474x + x2 = 1.684210526 + 0.09972299170

Combine like terms: 1.684210526 + 0.09972299170 = 1.7839335177
0.09972299170 + 0.6315789474x + x2 = 1.7839335177

Factor a perfect square on the left side:
(x + 0.3157894737)(x + 0.3157894737) = 1.7839335177

Calculate the square root of the right side: 1.335639741

Break this problem into two subproblems by setting 
(x + 0.3157894737) equal to 1.335639741 and -1.335639741.

Subproblem 1

x + 0.3157894737 = 1.335639741 Simplifying x + 0.3157894737 = 1.335639741 Reorder the terms: 0.3157894737 + x = 1.335639741 Solving 0.3157894737 + x = 1.335639741 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3157894737' to each side of the equation. 0.3157894737 + -0.3157894737 + x = 1.335639741 + -0.3157894737 Combine like terms: 0.3157894737 + -0.3157894737 = 0.0000000000 0.0000000000 + x = 1.335639741 + -0.3157894737 x = 1.335639741 + -0.3157894737 Combine like terms: 1.335639741 + -0.3157894737 = 1.0198502673 x = 1.0198502673 Simplifying x = 1.0198502673

Subproblem 2

x + 0.3157894737 = -1.335639741 Simplifying x + 0.3157894737 = -1.335639741 Reorder the terms: 0.3157894737 + x = -1.335639741 Solving 0.3157894737 + x = -1.335639741 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3157894737' to each side of the equation. 0.3157894737 + -0.3157894737 + x = -1.335639741 + -0.3157894737 Combine like terms: 0.3157894737 + -0.3157894737 = 0.0000000000 0.0000000000 + x = -1.335639741 + -0.3157894737 x = -1.335639741 + -0.3157894737 Combine like terms: -1.335639741 + -0.3157894737 = -1.6514292147 x = -1.6514292147 Simplifying x = -1.6514292147

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.0198502673, -1.6514292147}

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