Precalculus Examples

$y = 3 x^{2} + 5 x + 8$

Rewrite the equation in vertex form.

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Complete the square for $3 x^{2} + 5 x + 8$ .

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

$a = 3$

$b = 5$

$c = 8$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{5}{2 \cdot 3}$

Multiply $2$ by $3$ .

$d = \frac{5}{6}$

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

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

$e = 8 - \frac{5^{2}}{4 \cdot 3}$

Simplify the right side.

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Simplify each term.

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Raise $5$ to the power of $2$ .

$e = 8 - \frac{25}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 8 - \frac{25}{12}$

To write $8$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$e = 8 \cdot \frac{12}{12} - \frac{25}{12}$

Combine $8$ and $\frac{12}{12}$ .

$e = \frac{8 \cdot 12}{12} - \frac{25}{12}$

Combine the numerators over the common denominator.

$e = \frac{8 \cdot 12 - 25}{12}$

Simplify the numerator.

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Multiply $8$ by $12$ .

$e = \frac{96 - 25}{12}$

Subtract $25$ from $96$ .

$e = \frac{71}{12}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x + \frac{5}{6})}^{2} + \frac{71}{12}$ .

$3 {(x + \frac{5}{6})}^{2} + \frac{71}{12}$

Set $y$ equal to the new right side.

$y = 3 {(x + \frac{5}{6})}^{2} + \frac{71}{12}$

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

$a = 3$

$h = - \frac{5}{6}$

$k = \frac{71}{12}$

Find the vertex $(h, k)$ .

$(- \frac{5}{6}, \frac{71}{12})$