Algebra Examples

$- \frac{3}{4} x^{2} + 9 x - 18$

Step 1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Combine $x^{2}$ and $\frac{3}{4}$ .

$y = - \frac{x^{2} \cdot 3}{4} + 9 x - 18$

Step 1.1.2

Move $3$ to the left of $x^{2}$ .

$y = - \frac{3 x^{2}}{4} + 9 x - 18$

Step 1.2

Complete the square for $- \frac{3 x^{2}}{4} + 9 x - 18$ .

Tap for more steps...

Step 1.2.1

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

$a = - \frac{3}{4}$

$b = 9$

$c = - 18$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

Tap for more steps...

Step 1.2.3.1

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

$d = \frac{9}{2 (- \frac{3}{4})}$

Step 1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.1

Simplify the denominator.

Tap for more steps...

Step 1.2.3.2.1.1

Multiply $- 1$ by $2$ .

$d = \frac{9}{- 2 (\frac{3}{4})}$

Step 1.2.3.2.1.2

Combine $- 2$ and $\frac{3}{4}$ .

$d = \frac{9}{\frac{- 2 \cdot 3}{4}}$

Step 1.2.3.2.2

Multiply $- 2$ by $3$ .

$d = \frac{9}{\frac{- 6}{4}}$

Step 1.2.3.2.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 1.2.3.2.3.1

Cancel the common factor of $- 6$ and $4$ .

Tap for more steps...

Step 1.2.3.2.3.1.1

Factor $2$ out of $- 6$ .

$d = \frac{9}{\frac{2 (- 3)}{4}}$

Step 1.2.3.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.2.3.1.2.1

Factor $2$ out of $4$ .

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

Step 1.2.3.2.3.1.2.2

Cancel the common factor.

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

Step 1.2.3.2.3.1.2.3

Rewrite the expression.

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

Step 1.2.3.2.3.2

Move the negative in front of the fraction.

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

Step 1.2.3.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = 9 (- \frac{2}{3})$

Step 1.2.3.2.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.3.2.5.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$d = 9 (\frac{- 2}{3})$

Step 1.2.3.2.5.2

Factor $3$ out of $9$ .

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

Step 1.2.3.2.5.3

Cancel the common factor.

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

Step 1.2.3.2.5.4

Rewrite the expression.

$d = 3 \cdot - 2$

Step 1.2.3.2.6

Multiply $3$ by $- 2$ .

$d = - 6$

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

$e = - 18 - \frac{9^{2}}{4 (- \frac{3}{4})}$

Step 1.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.1

Simplify each term.

Tap for more steps...

Step 1.2.4.2.1.1

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

$e = - 18 - \frac{81}{4 (- \frac{3}{4})}$

Step 1.2.4.2.1.2

Simplify the denominator.

Tap for more steps...

Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - 18 - \frac{81}{- 4 (\frac{3}{4})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{3}{4}$ .

$e = - 18 - \frac{81}{\frac{- 4 \cdot 3}{4}}$

Step 1.2.4.2.1.3

Multiply $- 4$ by $3$ .

$e = - 18 - \frac{81}{\frac{- 12}{4}}$

Step 1.2.4.2.1.4

Divide $- 12$ by $4$ .

$e = - 18 - \frac{81}{- 3}$

Step 1.2.4.2.1.5

Divide $81$ by $- 3$ .

$e = - 18 - - 27$

Step 1.2.4.2.1.6

Multiply $- 1$ by $- 27$ .

$e = - 18 + 27$

Step 1.2.4.2.2

Add $- 18$ and $27$ .

$e = 9$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{3}{4} {(x - 6)}^{2} + 9$ .

$- \frac{3}{4} {(x - 6)}^{2} + 9$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{3}{4} \cdot {(x - 6)}^{2} + 9$

Step 2

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

$a = - \frac{3}{4}$

$h = 6$

$k = 9$

Step 3

Find the vertex $(h, k)$ .

$(6, 9)$

Step 4