Find Where Undefined/Discontinuous (9x^2)/(36x^3+27x^2)

$\frac{9 x^{2}}{36 x^{3} + 27 x^{2}}$

Step 1

Set the denominator in $\frac{9 x^{2}}{36 x^{3} + 27 x^{2}}$ equal to $0$ to find where the expression is undefined.

$36 x^{3} + 27 x^{2} = 0$

Step 2

Solve for $x$ .

Factor $9 x^{2}$ out of $36 x^{3} + 27 x^{2}$ .

Factor $9 x^{2}$ out of $36 x^{3}$ .

$9 x^{2} (4 x) + 27 x^{2} = 0$

Factor $9 x^{2}$ out of $27 x^{2}$ .

$9 x^{2} (4 x) + 9 x^{2} (3) = 0$

Factor $9 x^{2}$ out of $9 x^{2} (4 x) + 9 x^{2} (3)$ .

$9 x^{2} (4 x + 3) = 0$

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

$x^{2} = 0$

$4 x + 3 = 0$

Set $x^{2}$ equal to $0$ and solve for $x$ .

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Solve $x^{2} = 0$ for $x$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Set $4 x + 3$ equal to $0$ and solve for $x$ .

Set $4 x + 3$ equal to $0$ .

$4 x + 3 = 0$

Solve $4 x + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$4 x = - 3$

Divide each term in $4 x = - 3$ by $4$ and simplify.

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

The final solution is all the values that make $9 x^{2} (4 x + 3) = 0$ true.

$x = 0, - \frac{3}{4}$

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 $0$ .

$x = - \frac{3}{4}, x = 0$

Step 4

Find Where Undefined/Discontinuous (9x^2)/(36x^3+27x^2)

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