Find the x and y Intercepts f(x)=(5x-10)/(2x^2+3x-14)

$f (x) = \frac{5 x - 10}{2 x^{2} + 3 x - 14}$

Step 1

Find the x-intercepts.

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

$0 = \frac{5 x - 10}{2 x^{2} + 3 x - 14}$

Solve the equation.

Set the numerator equal to zero.

$5 x - 10 = 0$

Solve the equation for $x$ .

Add $10$ to both sides of the equation.

$5 x = 10$

Divide each term in $5 x = 10$ by $5$ and simplify.

Divide each term in $5 x = 10$ by $5$ .

$\frac{5 x}{5} = \frac{10}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{10}{5}$

Divide $x$ by $1$ .

$x = \frac{10}{5}$

Simplify the right side.

Divide $10$ by $5$ .

$x = 2$

Exclude the solutions that do not make $0 = \frac{5 x - 10}{2 x^{2} + 3 x - 14}$ true.

$No solution$

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.

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

$y = \frac{5 (0) - 10}{2 {(0)}^{2} + 3 (0) - 14}$

Solve the equation.

Remove parentheses.

$y = \frac{5 (0) - 10}{2 \cdot 0^{2} + 3 (0) - 14}$

Remove parentheses.

$y = \frac{5 (0) - 10}{2 {(0)}^{2} + 3 (0) - 14}$

Simplify $\frac{5 (0) - 10}{2 {(0)}^{2} + 3 (0) - 14}$ .

Cancel the common factor of $5 (0) - 10$ and $2 {(0)}^{2} + 3 (0) - 14$ .

Factor $2$ out of $5 (0)$ .

$y = \frac{2 (5 \cdot (0)) - 10}{2 {(0)}^{2} + 3 (0) - 14}$

Factor $2$ out of $- 10$ .

$y = \frac{2 (5 \cdot (0)) + 2 (- 5)}{2 {(0)}^{2} + 3 (0) - 14}$

Factor $2$ out of $2 (5 \cdot (0)) + 2 (- 5)$ .

$y = \frac{2 (5 \cdot (0) - 5)}{2 {(0)}^{2} + 3 (0) - 14}$

Cancel the common factors.

Factor $2$ out of $2 {(0)}^{2}$ .

$y = \frac{2 (5 \cdot (0) - 5)}{2 ({(0)}^{2}) + 3 (0) - 14}$

Factor $2$ out of $3 (0)$ .

$y = \frac{2 (5 \cdot (0) - 5)}{2 ({(0)}^{2}) + 2 (3 \cdot (0)) - 14}$

Factor $2$ out of $2 ({(0)}^{2}) + 2 (3 \cdot (0))$ .

$y = \frac{2 (5 \cdot (0) - 5)}{2 ({(0)}^{2} + 3 \cdot (0)) - 14}$

Factor $2$ out of $- 14$ .

$y = \frac{2 (5 \cdot (0) - 5)}{2 ({(0)}^{2} + 3 \cdot (0)) + 2 (- 7)}$

Factor $2$ out of $2 ({(0)}^{2} + 3 \cdot (0)) + 2 (- 7)$ .

$y = \frac{2 (5 \cdot (0) - 5)}{2 ({(0)}^{2} + 3 \cdot (0) - 7)}$

Cancel the common factor.

$y = \frac{2 (5 \cdot (0) - 5)}{2 ({(0)}^{2} + 3 \cdot (0) - 7)}$

Rewrite the expression.

$y = \frac{5 \cdot (0) - 5}{{(0)}^{2} + 3 \cdot (0) - 7}$

Simplify the numerator.

Multiply $5$ by $0$ .

$y = \frac{0 - 5}{0^{2} + 3 \cdot (0) - 7}$

Subtract $5$ from $0$ .

$y = \frac{- 5}{0^{2} + 3 \cdot (0) - 7}$

Simplify the denominator.

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

$y = \frac{- 5}{0 + 3 \cdot (0) - 7}$

Multiply $3$ by $0$ .

$y = \frac{- 5}{0 + 0 - 7}$

Add $0$ and $0$ .

$y = \frac{- 5}{0 - 7}$

Subtract $7$ from $0$ .

$y = \frac{- 5}{- 7}$

Dividing two negative values results in a positive value.

$y = \frac{5}{7}$

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{5}{7})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, \frac{5}{7})$

Step 4

Find the x and y Intercepts f(x)=(5x-10)/(2x^2+3x-14)

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