Precalculus Examples

$f (x) = 3 x^{2} + 5 x - 4$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $3 x^{2} + 5 x - 4$ .

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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 $c$ .

$a = 3$

$b = 5$

$c = - 4$

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 $d = \frac{b}{2 a}$ .

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

Step 1.1.3.2

Multiply $2$ by $3$ .

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

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 $e = c - \frac{b^{2}}{4 a}$ .

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

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 $5$ to the power of $2$ .

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

Step 1.1.4.2.1.2

Multiply $4$ by $3$ .

$e = - 4 - \frac{25}{12}$

Step 1.1.4.2.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$e = - 4 \cdot \frac{12}{12} - \frac{25}{12}$

Step 1.1.4.2.3

Combine $- 4$ and $\frac{12}{12}$ .

$e = \frac{- 4 \cdot 12}{12} - \frac{25}{12}$

Step 1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 4 \cdot 12 - 25}{12}$

Step 1.1.4.2.5

Simplify the numerator.

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Step 1.1.4.2.5.1

Multiply $- 4$ by $12$ .

$e = \frac{- 48 - 25}{12}$

Step 1.1.4.2.5.2

Subtract $25$ from $- 48$ .

$e = \frac{- 73}{12}$

Step 1.1.4.2.6

Move the negative in front of the fraction.

$e = - \frac{73}{12}$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x + \frac{5}{6})}^{2} - \frac{73}{12}$ .

$3 {(x + \frac{5}{6})}^{2} - \frac{73}{12}$

Step 1.2

Set $y$ equal to the new right side.

$y = 3 {(x + \frac{5}{6})}^{2} - \frac{73}{12}$

Step 2

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

$a = 3$

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

$k = - \frac{73}{12}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{5}{6}, - \frac{73}{12})$

Step 4