0=x^2-9x-32

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Solution for 0=x^2-9x-32 equation:


Simplifying
0 = x2 + -9x + -32

Reorder the terms:
0 = -32 + -9x + x2

Solving
0 = -32 + -9x + x2

Solving for variable 'x'.

Combine like terms: 0 + 32 = 32
32 + 9x + -1x2 = -32 + -9x + x2 + 32 + 9x + -1x2

Reorder the terms:
32 + 9x + -1x2 = -32 + 32 + -9x + 9x + x2 + -1x2

Combine like terms: -32 + 32 = 0
32 + 9x + -1x2 = 0 + -9x + 9x + x2 + -1x2
32 + 9x + -1x2 = -9x + 9x + x2 + -1x2

Combine like terms: -9x + 9x = 0
32 + 9x + -1x2 = 0 + x2 + -1x2
32 + 9x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
32 + 9x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-32 + -9x + x2 = 0

Move the constant term to the right:

Add '32' to each side of the equation.
-32 + -9x + 32 + x2 = 0 + 32

Reorder the terms:
-32 + 32 + -9x + x2 = 0 + 32

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

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

The x term is -9x.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9x + 20.25 + x2 = 32 + 20.25

Reorder the terms:
20.25 + -9x + x2 = 32 + 20.25

Combine like terms: 32 + 20.25 = 52.25
20.25 + -9x + x2 = 52.25

Factor a perfect square on the left side:
(x + -4.5)(x + -4.5) = 52.25

Calculate the square root of the right side: 7.228416147

Break this problem into two subproblems by setting 
(x + -4.5) equal to 7.228416147 and -7.228416147.

Subproblem 1

x + -4.5 = 7.228416147 Simplifying x + -4.5 = 7.228416147 Reorder the terms: -4.5 + x = 7.228416147 Solving -4.5 + x = 7.228416147 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = 7.228416147 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = 7.228416147 + 4.5 x = 7.228416147 + 4.5 Combine like terms: 7.228416147 + 4.5 = 11.728416147 x = 11.728416147 Simplifying x = 11.728416147

Subproblem 2

x + -4.5 = -7.228416147 Simplifying x + -4.5 = -7.228416147 Reorder the terms: -4.5 + x = -7.228416147 Solving -4.5 + x = -7.228416147 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = -7.228416147 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = -7.228416147 + 4.5 x = -7.228416147 + 4.5 Combine like terms: -7.228416147 + 4.5 = -2.728416147 x = -2.728416147 Simplifying x = -2.728416147

Solution

The solution to the problem is based on the solutions from the subproblems. x = {11.728416147, -2.728416147}

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