Precalculus Examples

$H (x) = \frac{3 x - 21}{49 - x^{2}}$

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 $x$ .

$0 = \frac{3 x - 21}{49 - x^{2}}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$3 x - 21 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Add $21$ to both sides of the equation.

$3 x = 21$

Step 1.2.2.2

Divide each term in $3 x = 21$ by $3$ and simplify.

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Step 1.2.2.2.1

Divide each term in $3 x = 21$ by $3$ .

$\frac{3 x}{3} = \frac{21}{3}$

Step 1.2.2.2.2

Simplify the left side.

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Step 1.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{21}{3}$

Step 1.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{21}{3}$

Step 1.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.3.1

Divide $21$ by $3$ .

$x = 7$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{3 x - 21}{49 - x^{2}}$ 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 $y$ .

$y = \frac{3 (0) - 21}{49 - {(0)}^{2}}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{3 (0) - 21}{49 - 0^{2}}$

Step 2.2.2

Remove parentheses.

$y = \frac{3 (0) - 21}{49 - {(0)}^{2}}$

Step 2.2.3

Simplify $\frac{3 (0) - 21}{49 - {(0)}^{2}}$ .

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Step 2.2.3.1

Simplify the numerator.

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Step 2.2.3.1.1

Multiply $3$ by $0$ .

$y = \frac{0 - 21}{49 - 0^{2}}$

Step 2.2.3.1.2

Subtract $21$ from $0$ .

$y = \frac{- 21}{49 - 0^{2}}$

Step 2.2.3.2

Simplify the denominator.

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Step 2.2.3.2.1

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

$y = \frac{- 21}{49 - 0}$

Step 2.2.3.2.2

Multiply $- 1$ by $0$ .

$y = \frac{- 21}{49 + 0}$

Step 2.2.3.2.3

Add $49$ and $0$ .

$y = \frac{- 21}{49}$

Step 2.2.3.3

Reduce the expression by cancelling the common factors.

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Step 2.2.3.3.1

Cancel the common factor of $- 21$ and $49$ .

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Step 2.2.3.3.1.1

Factor $7$ out of $- 21$ .

$y = \frac{7 (- 3)}{49}$

Step 2.2.3.3.1.2

Cancel the common factors.

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Step 2.2.3.3.1.2.1

Factor $7$ out of $49$ .

$y = \frac{7 \cdot - 3}{7 \cdot 7}$

Step 2.2.3.3.1.2.2

Cancel the common factor.

$y = \frac{7 \cdot - 3}{7 \cdot 7}$

Step 2.2.3.3.1.2.3

Rewrite the expression.

$y = \frac{- 3}{7}$

Step 2.2.3.3.2

Move the negative in front of the fraction.

$y = - \frac{3}{7}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{3}{7})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{3}{7})$

Step 4