Find the x and y Intercepts f(x)=3x^2+11x+5

$f (x) = 3 x^{2} + 11 x + 5$

Step 1

Find the x-intercepts.

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

$0 = 3 x^{2} + 11 x + 5$

Solve the equation.

Rewrite the equation as $3 x^{2} + 11 x + 5 = 0$ .

$3 x^{2} + 11 x + 5 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 3$ , $b = 11$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{- 11 \pm \sqrt{11^{2} - 4 \cdot (3 \cdot 5)}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Raise $11$ to the power of $2$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 5$ .

Multiply $- 4$ by $3$ .

$x = \frac{- 11 \pm \sqrt{121 - 12 \cdot 5}}{2 \cdot 3}$

Multiply $- 12$ by $5$ .

$x = \frac{- 11 \pm \sqrt{121 - 60}}{2 \cdot 3}$

Subtract $60$ from $121$ .

$x = \frac{- 11 \pm \sqrt{61}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 11 \pm \sqrt{61}}{6}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $11$ to the power of $2$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 5$ .

Multiply $- 4$ by $3$ .

$x = \frac{- 11 \pm \sqrt{121 - 12 \cdot 5}}{2 \cdot 3}$

Multiply $- 12$ by $5$ .

$x = \frac{- 11 \pm \sqrt{121 - 60}}{2 \cdot 3}$

Subtract $60$ from $121$ .

$x = \frac{- 11 \pm \sqrt{61}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 11 \pm \sqrt{61}}{6}$

Change the $\pm$ to $+$ .

$x = \frac{- 11 + \sqrt{61}}{6}$

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 1 \cdot 11 + \sqrt{61}}{6}$

Factor $- 1$ out of $\sqrt{61}$ .

$x = \frac{- 1 \cdot 11 - 1 (- \sqrt{61})}{6}$

Factor $- 1$ out of $- 1 (11) - 1 (- \sqrt{61})$ .

$x = \frac{- 1 (11 - \sqrt{61})}{6}$

Move the negative in front of the fraction.

$x = - \frac{11 - \sqrt{61}}{6}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $11$ to the power of $2$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 5$ .

Multiply $- 4$ by $3$ .

$x = \frac{- 11 \pm \sqrt{121 - 12 \cdot 5}}{2 \cdot 3}$

Multiply $- 12$ by $5$ .

$x = \frac{- 11 \pm \sqrt{121 - 60}}{2 \cdot 3}$

Subtract $60$ from $121$ .

$x = \frac{- 11 \pm \sqrt{61}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 11 \pm \sqrt{61}}{6}$

Change the $\pm$ to $-$ .

$x = \frac{- 11 - \sqrt{61}}{6}$

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 1 \cdot 11 - \sqrt{61}}{6}$

Factor $- 1$ out of $- \sqrt{61}$ .

$x = \frac{- 1 \cdot 11 - (\sqrt{61})}{6}$

Factor $- 1$ out of $- 1 (11) - (\sqrt{61})$ .

$x = \frac{- 1 (11 + \sqrt{61})}{6}$

Move the negative in front of the fraction.

$x = - \frac{11 + \sqrt{61}}{6}$

The final answer is the combination of both solutions.

$x = - \frac{11 - \sqrt{61}}{6}, - \frac{11 + \sqrt{61}}{6}$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{11 - \sqrt{61}}{6}, 0), (- \frac{11 + \sqrt{61}}{6}, 0)$

Step 2

Find the y-intercepts.

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

$y = 3 {(0)}^{2} + 11 (0) + 5$

Solve the equation.

Remove parentheses.

$y = 3 \cdot 0^{2} + 11 (0) + 5$

Remove parentheses.

$y = 3 {(0)}^{2} + 11 (0) + 5$

Simplify $3 {(0)}^{2} + 11 (0) + 5$ .

Simplify each term.

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

$y = 3 \cdot 0 + 11 (0) + 5$

Multiply $3$ by $0$ .

$y = 0 + 11 (0) + 5$

Multiply $11$ by $0$ .

$y = 0 + 0 + 5$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 5$

Add $0$ and $5$ .

$y = 5$

y-intercept(s) in point form.

y-intercept(s): $(0, 5)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{11 - \sqrt{61}}{6}, 0), (- \frac{11 + \sqrt{61}}{6}, 0)$

y-intercept(s): $(0, 5)$

Step 4

Find the x and y Intercepts f(x)=3x^2+11x+5

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