Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Use the quadratic formula to find the solutions.

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

Step 1.2.3

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

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

Step 1.2.4

Simplify.

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Step 1.2.4.1

Simplify the numerator.

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Step 1.2.4.1.1

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

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

Step 1.2.4.1.2

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

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Step 1.2.4.1.2.1

Multiply $- 4$ by $3$ .

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

Step 1.2.4.1.2.2

Multiply $- 12$ by $5$ .

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

Step 1.2.4.1.3

Subtract $60$ from $121$ .

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

Step 1.2.4.2

Multiply $2$ by $3$ .

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

Step 1.2.5

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

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

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

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

Step 1.2.5.1.2

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

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Step 1.2.5.1.2.1

Multiply $- 4$ by $3$ .

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

Step 1.2.5.1.2.2

Multiply $- 12$ by $5$ .

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

Step 1.2.5.1.3

Subtract $60$ from $121$ .

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

Step 1.2.5.2

Multiply $2$ by $3$ .

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

Step 1.2.5.3

Change the $\pm$ to $+$ .

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

Step 1.2.5.4

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

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

Step 1.2.5.5

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

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

Step 1.2.5.6

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

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

Step 1.2.5.7

Move the negative in front of the fraction.

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

Step 1.2.6

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

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

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

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

Step 1.2.6.1.2

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

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Step 1.2.6.1.2.1

Multiply $- 4$ by $3$ .

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

Step 1.2.6.1.2.2

Multiply $- 12$ by $5$ .

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

Step 1.2.6.1.3

Subtract $60$ from $121$ .

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

Step 1.2.6.2

Multiply $2$ by $3$ .

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

Step 1.2.6.3

Change the $\pm$ to $-$ .

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

Step 1.2.6.4

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

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

Step 1.2.6.5

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

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

Step 1.2.6.6

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

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

Step 1.2.6.7

Move the negative in front of the fraction.

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

Step 1.2.7

The final answer is the combination of both solutions.

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

Step 1.3

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.

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Step 2.1

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

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

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

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

Step 2.2.3.1.2

Multiply $3$ by $0$ .

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

Step 2.2.3.1.3

Multiply $11$ by $0$ .

$y = 0 + 0 + 5$

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 5$

Step 2.2.3.2.2

Add $0$ and $5$ .

$y = 5$

Step 2.3

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