Algebra Examples

$\frac{9 k^{2} - 33 k + 10}{18 k^{3} - 200 k}$

Step 1

Set the denominator in $\frac{9 k^{2} - 33 k + 10}{18 k^{3} - 200 k}$ equal to $0$ to find where the expression is undefined.

$18 k^{3} - 200 k = 0$

Step 2

Solve for $k$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $2 k$ out of $18 k^{3} - 200 k$ .

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Step 2.1.1.1

Factor $2 k$ out of $18 k^{3}$ .

$2 k (9 k^{2}) - 200 k = 0$

Step 2.1.1.2

Factor $2 k$ out of $- 200 k$ .

$2 k (9 k^{2}) + 2 k (- 100) = 0$

Step 2.1.1.3

Factor $2 k$ out of $2 k (9 k^{2}) + 2 k (- 100)$ .

$2 k (9 k^{2} - 100) = 0$

Step 2.1.2

Rewrite $9 k^{2}$ as ${(3 k)}^{2}$ .

$2 k ({(3 k)}^{2} - 100) = 0$

Step 2.1.3

Rewrite $100$ as $10^{2}$ .

$2 k ({(3 k)}^{2} - 10^{2}) = 0$

Step 2.1.4

Factor.

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Step 2.1.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 k$ and $b = 10$ .

$2 k ((3 k + 10) (3 k - 10)) = 0$

Step 2.1.4.2

Remove unnecessary parentheses.

$2 k (3 k + 10) (3 k - 10) = 0$

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 $0$ .

$k = 0$

$3 k + 10 = 0$

$3 k - 10 = 0$

Step 2.3

Set $k$ equal to $0$ .

$k = 0$

Step 2.4

Set $3 k + 10$ equal to $0$ and solve for $k$ .

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Step 2.4.1

Set $3 k + 10$ equal to $0$ .

$3 k + 10 = 0$

Step 2.4.2

Solve $3 k + 10 = 0$ for $k$ .

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Step 2.4.2.1

Subtract $10$ from both sides of the equation.

$3 k = - 10$

Step 2.4.2.2

Divide each term in $3 k = - 10$ by $3$ and simplify.

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Step 2.4.2.2.1

Divide each term in $3 k = - 10$ by $3$ .

$\frac{3 k}{3} = \frac{- 10}{3}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 k}{3} = \frac{- 10}{3}$

Step 2.4.2.2.2.1.2

Divide $k$ by $1$ .

$k = \frac{- 10}{3}$

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Move the negative in front of the fraction.

$k = - \frac{10}{3}$

Step 2.5

Set $3 k - 10$ equal to $0$ and solve for $k$ .

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Step 2.5.1

Set $3 k - 10$ equal to $0$ .

$3 k - 10 = 0$

Step 2.5.2

Solve $3 k - 10 = 0$ for $k$ .

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Step 2.5.2.1

Add $10$ to both sides of the equation.

$3 k = 10$

Step 2.5.2.2

Divide each term in $3 k = 10$ by $3$ and simplify.

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Step 2.5.2.2.1

Divide each term in $3 k = 10$ by $3$ .

$\frac{3 k}{3} = \frac{10}{3}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 k}{3} = \frac{10}{3}$

Step 2.5.2.2.2.1.2

Divide $k$ by $1$ .

$k = \frac{10}{3}$

Step 2.6

The final solution is all the values that make $2 k (3 k + 10) (3 k - 10) = 0$ true.

$k = 0, - \frac{10}{3}, \frac{10}{3}$

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 $0$ .

$k = - \frac{10}{3}, k = 0, k = \frac{10}{3}$

Step 4