Algebra Examples

$y = \frac{(x - 3)}{x^{2} - 12 x + 27}$

Step 1

Set the denominator in $\frac{x - 3}{x^{2} - 12 x + 27}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 12 x + 27 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x^{2} - 12 x + 27$ using the AC method.

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Step 2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $27$ and whose sum is $- 12$ .

$- 9, - 3$

Step 2.1.2

Write the factored form using these integers.

$(x - 9) (x - 3) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 9 = 0$

$x - 3 = 0$

Step 2.3

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 2.3.2

Add $9$ to both sides of the equation.

$x = 9$

Step 2.4

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.5

The final solution is all the values that make $(x - 9) (x - 3) = 0$ true.

$x = 9, 3$

Step 3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = 3, x = 9$

Step 4