Algebra Examples

$y = 3 x^{2} - 9 x + 12$

Step 1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1

Complete the square for $3 x^{2} - 9 x + 12$ .

Tap for more steps...

Step 1.1.1

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

$a = 3$

$b = - 9$

$c = 12$

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

Tap for more steps...

Step 1.1.3.1

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

$d = \frac{- 9}{2 \cdot 3}$

Step 1.1.3.2

Simplify the right side.

Tap for more steps...

Step 1.1.3.2.1

Cancel the common factor of $- 9$ and $3$ .

Tap for more steps...

Step 1.1.3.2.1.1

Factor $3$ out of $- 9$ .

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

Step 1.1.3.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.1.3.2.1.2.1

Factor $3$ out of $2 \cdot 3$ .

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

Step 1.1.3.2.1.2.2

Cancel the common factor.

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

Step 1.1.3.2.1.2.3

Rewrite the expression.

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

Step 1.1.3.2.2

Move the negative in front of the fraction.

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

Step 1.1.4

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

Tap for more steps...

Step 1.1.4.1

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

$e = 12 - \frac{{(- 9)}^{2}}{4 \cdot 3}$

Step 1.1.4.2

Simplify the right side.

Tap for more steps...

Step 1.1.4.2.1

Simplify each term.

Tap for more steps...

Step 1.1.4.2.1.1

Raise $- 9$ to the power of $2$ .

$e = 12 - \frac{81}{4 \cdot 3}$

Step 1.1.4.2.1.2

Multiply $4$ by $3$ .

$e = 12 - \frac{81}{12}$

Step 1.1.4.2.1.3

Cancel the common factor of $81$ and $12$ .

Tap for more steps...

Step 1.1.4.2.1.3.1

Factor $3$ out of $81$ .

$e = 12 - \frac{3 (27)}{12}$

Step 1.1.4.2.1.3.2

Cancel the common factors.

Tap for more steps...

Step 1.1.4.2.1.3.2.1

Factor $3$ out of $12$ .

$e = 12 - \frac{3 \cdot 27}{3 \cdot 4}$

Step 1.1.4.2.1.3.2.2

Cancel the common factor.

$e = 12 - \frac{3 \cdot 27}{3 \cdot 4}$

Step 1.1.4.2.1.3.2.3

Rewrite the expression.

$e = 12 - \frac{27}{4}$

Step 1.1.4.2.2

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

$e = 12 \cdot \frac{4}{4} - \frac{27}{4}$

Step 1.1.4.2.3

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

$e = \frac{12 \cdot 4}{4} - \frac{27}{4}$

Step 1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{12 \cdot 4 - 27}{4}$

Step 1.1.4.2.5

Simplify the numerator.

Tap for more steps...

Step 1.1.4.2.5.1

Multiply $12$ by $4$ .

$e = \frac{48 - 27}{4}$

Step 1.1.4.2.5.2

Subtract $27$ from $48$ .

$e = \frac{21}{4}$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x - \frac{3}{2})}^{2} + \frac{21}{4}$ .

$3 {(x - \frac{3}{2})}^{2} + \frac{21}{4}$

Step 1.2

Set $y$ equal to the new right side.

$y = 3 {(x - \frac{3}{2})}^{2} + \frac{21}{4}$

Step 2

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

$a = 3$

$h = \frac{3}{2}$

$k = \frac{21}{4}$

Step 3

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(\frac{3}{2}, \frac{21}{4})$

Step 5

Find $p$ , the distance from the vertex to the focus.

Tap for more steps...

Step 5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot 3}$

Step 5.3

Multiply $4$ by $3$ .

$\frac{1}{12}$

Step 6

Find the focus.

Tap for more steps...

Step 6.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 6.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(\frac{3}{2}, \frac{16}{3})$

Step 7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = \frac{3}{2}$

Step 8