9x^2-32x=19

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Solution for 9x^2-32x=19 equation:


Simplifying
9x2 + -32x = 19

Reorder the terms:
-32x + 9x2 = 19

Solving
-32x + 9x2 = 19

Solving for variable 'x'.

Reorder the terms:
-19 + -32x + 9x2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + -32x + 9x2 = 0

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

Divide each side by '9'.
-2.111111111 + -3.555555556x + x2 = 0

Move the constant term to the right:

Add '2.111111111' to each side of the equation.
-2.111111111 + -3.555555556x + 2.111111111 + x2 = 0 + 2.111111111

Reorder the terms:
-2.111111111 + 2.111111111 + -3.555555556x + x2 = 0 + 2.111111111

Combine like terms: -2.111111111 + 2.111111111 = 0.000000000
0.000000000 + -3.555555556x + x2 = 0 + 2.111111111
-3.555555556x + x2 = 0 + 2.111111111

Combine like terms: 0 + 2.111111111 = 2.111111111
-3.555555556x + x2 = 2.111111111

The x term is -3.555555556x.  Take half its coefficient (-1.777777778).
Square it (3.160493828) and add it to both sides.

Add '3.160493828' to each side of the equation.
-3.555555556x + 3.160493828 + x2 = 2.111111111 + 3.160493828

Reorder the terms:
3.160493828 + -3.555555556x + x2 = 2.111111111 + 3.160493828

Combine like terms: 2.111111111 + 3.160493828 = 5.271604939
3.160493828 + -3.555555556x + x2 = 5.271604939

Factor a perfect square on the left side:
(x + -1.777777778)(x + -1.777777778) = 5.271604939

Calculate the square root of the right side: 2.295997591

Break this problem into two subproblems by setting 
(x + -1.777777778) equal to 2.295997591 and -2.295997591.

Subproblem 1

x + -1.777777778 = 2.295997591 Simplifying x + -1.777777778 = 2.295997591 Reorder the terms: -1.777777778 + x = 2.295997591 Solving -1.777777778 + x = 2.295997591 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.777777778' to each side of the equation. -1.777777778 + 1.777777778 + x = 2.295997591 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + x = 2.295997591 + 1.777777778 x = 2.295997591 + 1.777777778 Combine like terms: 2.295997591 + 1.777777778 = 4.073775369 x = 4.073775369 Simplifying x = 4.073775369

Subproblem 2

x + -1.777777778 = -2.295997591 Simplifying x + -1.777777778 = -2.295997591 Reorder the terms: -1.777777778 + x = -2.295997591 Solving -1.777777778 + x = -2.295997591 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.777777778' to each side of the equation. -1.777777778 + 1.777777778 + x = -2.295997591 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + x = -2.295997591 + 1.777777778 x = -2.295997591 + 1.777777778 Combine like terms: -2.295997591 + 1.777777778 = -0.518219813 x = -0.518219813 Simplifying x = -0.518219813

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.073775369, -0.518219813}

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