Algebra Examples

$\frac{81 y^{2} - 169}{9 y^{2} - 22 y + 13}$

Step 1

Set the denominator in $\frac{81 y^{2} - 169}{9 y^{2} - 22 y + 13}$ equal to $0$ to find where the expression is undefined.

$9 y^{2} - 22 y + 13 = 0$

Step 2

Solve for $y$ .

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Step 2.1

Factor by grouping.

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Step 2.1.1

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 $a \cdot c = 9 \cdot 13 = 117$ and whose sum is $b = - 22$ .

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Step 2.1.1.1

Factor $- 22$ out of $- 22 y$ .

$9 y^{2} - 22 y + 13 = 0$

Step 2.1.1.2

Rewrite $- 22$ as $- 9$ plus $- 13$

$9 y^{2} + (- 9 - 13) y + 13 = 0$

Step 2.1.1.3

Apply the distributive property.

$9 y^{2} - 9 y - 13 y + 13 = 0$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$(9 y^{2} - 9 y) - 13 y + 13 = 0$

Step 2.1.2.2

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

$9 y (y - 1) - 13 (y - 1) = 0$

Step 2.1.3

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

$(y - 1) (9 y - 13) = 0$

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 $0$ .

$y - 1 = 0$

$9 y - 13 = 0$

Step 2.3

Set $y - 1$ equal to $0$ and solve for $y$ .

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Step 2.3.1

Set $y - 1$ equal to $0$ .

$y - 1 = 0$

Step 2.3.2

Add $1$ to both sides of the equation.

$y = 1$

Step 2.4

Set $9 y - 13$ equal to $0$ and solve for $y$ .

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Step 2.4.1

Set $9 y - 13$ equal to $0$ .

$9 y - 13 = 0$

Step 2.4.2

Solve $9 y - 13 = 0$ for $y$ .

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Step 2.4.2.1

Add $13$ to both sides of the equation.

$9 y = 13$

Step 2.4.2.2

Divide each term in $9 y = 13$ by $9$ and simplify.

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Step 2.4.2.2.1

Divide each term in $9 y = 13$ by $9$ .

$\frac{9 y}{9} = \frac{13}{9}$

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 $9$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{9 y}{9} = \frac{13}{9}$

Step 2.4.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{13}{9}$

Step 2.5

The final solution is all the values that make $(y - 1) (9 y - 13) = 0$ true.

$y = 1, \frac{13}{9}$

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 $0$ .

$y = 1, y = \frac{13}{9}$

Step 4