Algebra Examples

$f (x) = \frac{x^{2} - 18 x + 81}{x^{2} - 9 x}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{x^{2} - 18 x + 81}{x^{2} - 9 x}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x^{2} - 18 x + 81 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Factor using the perfect square rule.

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Step 1.2.2.1.1

Rewrite $81$ as $9^{2}$ .

$x^{2} - 18 x + 9^{2} = 0$

Step 1.2.2.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$18 x = 2 \cdot x \cdot 9$

Step 1.2.2.1.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 9 + 9^{2} = 0$

Step 1.2.2.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 9$ .

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

Step 1.2.2.2

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

$x - 9 = 0$

Step 1.2.2.3

Add $9$ to both sides of the equation.

$x = 9$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{x^{2} - 18 x + 81}{x^{2} - 9 x}$ true.

$No solution$

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{{(0)}^{2} - 18 \cdot 0 + 81}{{(0)}^{2} - 9 \cdot 0}$

Step 2.2

The equation has an undefined fraction.

Undefined

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $None$

Step 4