Algebra Examples

$f (x) = \frac{x - 10}{3 x^{2} - 51 x + 210}$

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 $x$ .

$0 = \frac{x - 10}{3 x^{2} - 51 x + 210}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x - 10 = 0$

Step 1.2.2

Add $10$ to both sides of the equation.

$x = 10$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{x - 10}{3 x^{2} - 51 x + 210}$ 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 $y$ .

$y = \frac{(0) - 10}{3 {(0)}^{2} - 51 \cdot 0 + 210}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{0 - 10}{3 {(0)}^{2} - 51 \cdot 0 + 210}$

Step 2.2.2

Remove parentheses.

$y = \frac{0 - 10}{3 \cdot 0^{2} - 51 \cdot 0 + 210}$

Step 2.2.3

Remove parentheses.

$y = \frac{(0) - 10}{3 {(0)}^{2} - 51 \cdot 0 + 210}$

Step 2.2.4

Simplify $\frac{(0) - 10}{3 {(0)}^{2} - 51 \cdot 0 + 210}$ .

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Step 2.2.4.1

Subtract $10$ from $0$ .

$y = \frac{- 10}{3 \cdot 0^{2} - 51 \cdot 0 + 210}$

Step 2.2.4.2

Simplify the denominator.

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Step 2.2.4.2.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{- 10}{3 \cdot 0 - 51 \cdot 0 + 210}$

Step 2.2.4.2.2

Multiply $3$ by $0$ .

$y = \frac{- 10}{0 - 51 \cdot 0 + 210}$

Step 2.2.4.2.3

Multiply $- 51$ by $0$ .

$y = \frac{- 10}{0 + 0 + 210}$

Step 2.2.4.2.4

Add $0$ and $0$ .

$y = \frac{- 10}{0 + 210}$

Step 2.2.4.2.5

Add $0$ and $210$ .

$y = \frac{- 10}{210}$

Step 2.2.4.3

Reduce the expression by cancelling the common factors.

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Step 2.2.4.3.1

Cancel the common factor of $- 10$ and $210$ .

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Step 2.2.4.3.1.1

Factor $10$ out of $- 10$ .

$y = \frac{10 (- 1)}{210}$

Step 2.2.4.3.1.2

Cancel the common factors.

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Step 2.2.4.3.1.2.1

Factor $10$ out of $210$ .

$y = \frac{10 \cdot - 1}{10 \cdot 21}$

Step 2.2.4.3.1.2.2

Cancel the common factor.

$y = \frac{10 \cdot - 1}{10 \cdot 21}$

Step 2.2.4.3.1.2.3

Rewrite the expression.

$y = \frac{- 1}{21}$

Step 2.2.4.3.2

Move the negative in front of the fraction.

$y = - \frac{1}{21}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{1}{21})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{1}{21})$

Step 4