Algebra Examples

$f (x) = \frac{2 x^{2} - 11 x + 9}{6 x^{2} - 27 x}$

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 $x$ .

$0 = \frac{2 x^{2} - 11 x + 9}{6 x^{2} - 27 x}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$2 x^{2} - 11 x + 9 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Factor by grouping.

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Step 1.2.2.1.1

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 $a \cdot c = 2 \cdot 9 = 18$ and whose sum is $b = - 11$ .

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Step 1.2.2.1.1.1

Factor $- 11$ out of $- 11 x$ .

$2 x^{2} - 11 x + 9 = 0$

Step 1.2.2.1.1.2

Rewrite $- 11$ as $- 2$ plus $- 9$

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

Step 1.2.2.1.1.3

Apply the distributive property.

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

Step 1.2.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2.1.2.2

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

$2 x (x - 1) - 9 (x - 1) = 0$

Step 1.2.2.1.3

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

$(x - 1) (2 x - 9) = 0$

Step 1.2.2.2

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

$x - 1 = 0$

$2 x - 9 = 0$

Step 1.2.2.3

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

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Step 1.2.2.3.1

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

$x - 1 = 0$

Step 1.2.2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.2.4

Set $2 x - 9$ equal to $0$ and solve for $x$ .

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Step 1.2.2.4.1

Set $2 x - 9$ equal to $0$ .

$2 x - 9 = 0$

Step 1.2.2.4.2

Solve $2 x - 9 = 0$ for $x$ .

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Step 1.2.2.4.2.1

Add $9$ to both sides of the equation.

$2 x = 9$

Step 1.2.2.4.2.2

Divide each term in $2 x = 9$ by $2$ and simplify.

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Step 1.2.2.4.2.2.1

Divide each term in $2 x = 9$ by $2$ .

$\frac{2 x}{2} = \frac{9}{2}$

Step 1.2.2.4.2.2.2

Simplify the left side.

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Step 1.2.2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{9}{2}$

Step 1.2.2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{2}$

Step 1.2.2.5

The final solution is all the values that make $(x - 1) (2 x - 9) = 0$ true.

$x = 1, \frac{9}{2}$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{2 x^{2} - 11 x + 9}{6 x^{2} - 27 x}$ true.

$x = 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(1, 0)$

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 $y$ .

$y = \frac{2 {(0)}^{2} - 11 \cdot 0 + 9}{6 {(0)}^{2} - 27 \cdot 0}$

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): $(1, 0)$

y-intercept(s): $None$

Step 4