Find the x and y Intercepts f(x)=3x^2+17x+20

$f (x) = 3 x^{2} + 17 x + 20$

Step 1

Find the x-intercepts.

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

$0 = 3 x^{2} + 17 x + 20$

Solve the equation.

Rewrite the equation as $3 x^{2} + 17 x + 20 = 0$ .

$3 x^{2} + 17 x + 20 = 0$

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 $a \cdot c = 3 \cdot 20 = 60$ and whose sum is $b = 17$ .

Factor $17$ out of $17 x$ .

$3 x^{2} + 17 (x) + 20 = 0$

Rewrite $17$ as $5$ plus $12$

$3 x^{2} + (5 + 12) x + 20 = 0$

Apply the distributive property.

$3 x^{2} + 5 x + 12 x + 20 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{2} + 5 x) + 12 x + 20 = 0$

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

$x (3 x + 5) + 4 (3 x + 5) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 5$ .

$(3 x + 5) (x + 4) = 0$

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

$3 x + 5 = 0$

$x + 4 = 0$

Set $3 x + 5$ equal to $0$ and solve for $x$ .

Set $3 x + 5$ equal to $0$ .

$3 x + 5 = 0$

Solve $3 x + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$3 x = - 5$

Divide each term in $3 x = - 5$ by $3$ and simplify.

Divide each term in $3 x = - 5$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(3 x + 5) (x + 4) = 0$ true.

$x = - \frac{5}{3}, - 4$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{5}{3}, 0), (- 4, 0)$

Step 2

Find the y-intercepts.

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

$y = 3 {(0)}^{2} + 17 (0) + 20$

Solve the equation.

Remove parentheses.

$y = 3 \cdot 0^{2} + 17 (0) + 20$

Remove parentheses.

$y = 3 {(0)}^{2} + 17 (0) + 20$

Simplify $3 {(0)}^{2} + 17 (0) + 20$ .

Simplify each term.

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

$y = 3 \cdot 0 + 17 (0) + 20$

Multiply $3$ by $0$ .

$y = 0 + 17 (0) + 20$

Multiply $17$ by $0$ .

$y = 0 + 0 + 20$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 20$

Add $0$ and $20$ .

$y = 20$

y-intercept(s) in point form.

y-intercept(s): $(0, 20)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{5}{3}, 0), (- 4, 0)$

y-intercept(s): $(0, 20)$

Step 4

Find the x and y Intercepts f(x)=3x^2+17x+20

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