Find the x and y Intercepts f(x)=(5x^2+7x-24)/(10x-16)

$f (x) = \frac{5 x^{2} + 7 x - 24}{10 x - 16}$

Find the x-intercepts.

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

$0 = \frac{5 x^{2} + 7 x - 24}{10 x - 16}$

Solve the equation.

Set the numerator equal to zero.

$5 x^{2} + 7 x - 24 = 0$

Solve the equation for $x$ .

Factor by grouping.

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 $a \cdot c = 5 \cdot - 24 = - 120$ and whose sum is $b = 7$ .

Factor $7$ out of $7 x$ .

$5 x^{2} + 7 (x) - 24 = 0$

Rewrite $7$ as $- 8$ plus $15$

$5 x^{2} + (- 8 + 15) x - 24 = 0$

Apply the distributive property.

$5 x^{2} - 8 x + 15 x - 24 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(5 x^{2} - 8 x) + 15 x - 24 = 0$

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

$x (5 x - 8) + 3 (5 x - 8) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 x - 8$ .

$(5 x - 8) (x + 3) = 0$

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

$5 x - 8 = 0$

$x + 3 = 0$

Set $5 x - 8$ equal to $0$ and solve for $x$ .

Set $5 x - 8$ equal to $0$ .

$5 x - 8 = 0$

Solve $5 x - 8 = 0$ for $x$ .

Add $8$ to both sides of the equation.

$5 x = 8$

Divide each term in $5 x = 8$ by $5$ and simplify.

Divide each term in $5 x = 8$ by $5$ .

$\frac{5 x}{5} = \frac{8}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{8}{5}$

Divide $x$ by $1$ .

$x = \frac{8}{5}$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $(5 x - 8) (x + 3) = 0$ true.

$x = \frac{8}{5}, - 3$

Exclude the solutions that do not make $0 = \frac{5 x^{2} + 7 x - 24}{10 x - 16}$ true.

$x = - 3$

x-intercept(s) in point form.

x-intercept(s): $(- 3, 0)$

Find the y-intercepts.

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

$y = \frac{5 {(0)}^{2} + 7 (0) - 24}{10 (0) - 16}$

Solve the equation.

Remove parentheses.

$y = \frac{5 \cdot 0^{2} + 7 (0) - 24}{10 (0) - 16}$

Remove parentheses.

$y = \frac{5 {(0)}^{2} + 7 (0) - 24}{10 (0) - 16}$

Simplify $\frac{5 {(0)}^{2} + 7 (0) - 24}{10 (0) - 16}$ .

Simplify the numerator.

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

$y = \frac{5 \cdot 0 + 7 (0) - 24}{10 (0) - 16}$

Multiply $5$ by $0$ .

$y = \frac{0 + 7 (0) - 24}{10 (0) - 16}$

Multiply $7$ by $0$ .

$y = \frac{0 + 0 - 24}{10 (0) - 16}$

Add $0$ and $0$ .

$y = \frac{0 - 24}{10 (0) - 16}$

Subtract $24$ from $0$ .

$y = \frac{- 24}{10 (0) - 16}$

Simplify the denominator.

Multiply $10$ by $0$ .

$y = \frac{- 24}{0 - 16}$

Subtract $16$ from $0$ .

$y = \frac{- 24}{- 16}$

Cancel the common factor of $- 24$ and $- 16$ .

Factor $- 8$ out of $- 24$ .

$y = \frac{- 8 (3)}{- 16}$

Cancel the common factors.

Factor $- 8$ out of $- 16$ .

$y = \frac{- 8 \cdot 3}{- 8 \cdot 2}$

Cancel the common factor.

$y = \frac{- 8 \cdot 3}{- 8 \cdot 2}$

Rewrite the expression.

$y = \frac{3}{2}$

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{3}{2})$

List the intersections.

x-intercept(s): $(- 3, 0)$

y-intercept(s): $(0, \frac{3}{2})$

Find the x and y Intercepts f(x)=(5x^2+7x-24)/(10x-16)

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