Algebra Examples

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

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $- 3 x^{2} - 11 x - 5$ .

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Step 1.1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 3$

$b = - 11$

$c = - 5$

Step 1.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 11}{2 \cdot - 3}$

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Multiply $2$ by $- 3$ .

$d = \frac{- 11}{- 6}$

Step 1.1.3.2.2

Dividing two negative values results in a positive value.

$d = \frac{11}{6}$

Step 1.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - 5 - \frac{{(- 11)}^{2}}{4 \cdot - 3}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

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

$e = - 5 - \frac{121}{4 \cdot - 3}$

Step 1.1.4.2.1.2

Multiply $4$ by $- 3$ .

$e = - 5 - \frac{121}{- 12}$

Step 1.1.4.2.1.3

Move the negative in front of the fraction.

$e = - 5 - - \frac{121}{12}$

Step 1.1.4.2.1.4

Multiply $- - \frac{121}{12}$ .

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Step 1.1.4.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = - 5 + 1 (\frac{121}{12})$

Step 1.1.4.2.1.4.2

Multiply $\frac{121}{12}$ by $1$ .

$e = - 5 + \frac{121}{12}$

Step 1.1.4.2.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$e = - 5 \cdot \frac{12}{12} + \frac{121}{12}$

Step 1.1.4.2.3

Combine $- 5$ and $\frac{12}{12}$ .

$e = \frac{- 5 \cdot 12}{12} + \frac{121}{12}$

Step 1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 5 \cdot 12 + 121}{12}$

Step 1.1.4.2.5

Simplify the numerator.

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Step 1.1.4.2.5.1

Multiply $- 5$ by $12$ .

$e = \frac{- 60 + 121}{12}$

Step 1.1.4.2.5.2

Add $- 60$ and $121$ .

$e = \frac{61}{12}$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(x + \frac{11}{6})}^{2} + \frac{61}{12}$ .

$- 3 {(x + \frac{11}{6})}^{2} + \frac{61}{12}$

Step 1.2

Set $y$ equal to the new right side.

$y = - 3 {(x + \frac{11}{6})}^{2} + \frac{61}{12}$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 3$

$h = - \frac{11}{6}$

$k = \frac{61}{12}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{11}{6}, \frac{61}{12})$

Step 4