Find Where Undefined/Discontinuous (9x^2-81)/(3x^2-10x+3)

$\frac{9 x^{2} - 81}{3 x^{2} - 10 x + 3}$

Step 1

Set the denominator in $\frac{9 x^{2} - 81}{3 x^{2} - 10 x + 3}$ equal to $0$ to find where the expression is undefined.

$3 x^{2} - 10 x + 3 = 0$

Step 2

Solve for $x$ .

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 3 = 9$ and whose sum is $b = - 10$ .

Factor $- 10$ out of $- 10 x$ .

$3 x^{2} - 10 x + 3 = 0$

Rewrite $- 10$ as $- 1$ plus $- 9$

$3 x^{2} + (- 1 - 9) x + 3 = 0$

Apply the distributive property.

$3 x^{2} - 1 x - 9 x + 3 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{2} - 1 x) - 9 x + 3 = 0$

Factor out the greatest common factor (GCF) from each group.

$x (3 x - 1) - 3 (3 x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(3 x - 1) (x - 3) = 0$

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

$3 x - 1 = 0$

$x - 3 = 0$

Set $3 x - 1$ equal to $0$ and solve for $x$ .

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Solve $3 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$3 x = 1$

Divide each term in $3 x = 1$ by $3$ and simplify.

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{1}{3}$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $(3 x - 1) (x - 3) = 0$ true.

$x = \frac{1}{3}, 3$

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 $0$ .

$x = \frac{1}{3}, x = 3$

Step 4

Find Where Undefined/Discontinuous (9x^2-81)/(3x^2-10x+3)

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