Algebra Examples

$s^{2} - \frac{2}{3} s +$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Combine $s$ and $\frac{2}{3}$ .

$s^{2} - \frac{s \cdot 2}{3}$

Step 1.2

Move $2$ to the left of $s$ .

$s^{2} - \frac{2 s}{3}$

Step 2

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

$a = 1$

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

$c = 0$

Step 3

Consider the vertex form of a parabola.

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

Step 4

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

Tap for more steps...

Step 4.1

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

$d = \frac{- \frac{2}{3}}{2 \cdot 1}$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $- 1$ and $1$ .

Tap for more steps...

Step 4.2.1.1

Rewrite $- 1$ as $- 1 (1)$ .

$d = \frac{- 1 (1) \frac{2}{3}}{2 \cdot 1}$

Step 4.2.1.2

Cancel the common factor.

$d = \frac{- 1 \cdot 1 \frac{2}{3}}{2 \cdot 1}$

Step 4.2.1.3

Rewrite the expression.

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

Step 4.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{2}{3} \cdot \frac{1}{2}$

Step 4.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.3.1

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

$d = \frac{- 2}{3} \cdot \frac{1}{2}$

Step 4.2.3.2

Factor $2$ out of $- 2$ .

$d = \frac{2 (- 1)}{3} \cdot \frac{1}{2}$

Step 4.2.3.3

Cancel the common factor.

$d = \frac{2 \cdot - 1}{3} \cdot \frac{1}{2}$

Step 4.2.3.4

Rewrite the expression.

$d = \frac{- 1}{3}$

Step 4.2.4

Move the negative in front of the fraction.

$d = - \frac{1}{3}$

Step 5

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

Tap for more steps...

Step 5.1

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

$e = 0 - \frac{{(- \frac{2}{3})}^{2}}{4 \cdot 1}$

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Simplify the numerator.

Tap for more steps...

Step 5.2.1.1.1

Apply the product rule to $- \frac{2}{3}$ .

$e = 0 - \frac{{(- 1)}^{2} {(\frac{2}{3})}^{2}}{4 \cdot 1}$

Step 5.2.1.1.2

Raise $- 1$ to the power of $2$ .

$e = 0 - \frac{1 {(\frac{2}{3})}^{2}}{4 \cdot 1}$

Step 5.2.1.1.3

Apply the product rule to $\frac{2}{3}$ .

$e = 0 - \frac{1 \frac{2^{2}}{3^{2}}}{4 \cdot 1}$

Step 5.2.1.1.4

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

$e = 0 - \frac{1 \frac{4}{3^{2}}}{4 \cdot 1}$

Step 5.2.1.1.5

Raise $3$ to the power of $2$ .

$e = 0 - \frac{1 (\frac{4}{9})}{4 \cdot 1}$

Step 5.2.1.1.6

Multiply $\frac{4}{9}$ by $1$ .

$e = 0 - \frac{\frac{4}{9}}{4 \cdot 1}$

Step 5.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{\frac{4}{9}}{4}$

Step 5.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (\frac{4}{9} \cdot \frac{1}{4})$

Step 5.2.1.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.2.1.4.1

Cancel the common factor.

$e = 0 - (\frac{4}{9} \cdot \frac{1}{4})$

Step 5.2.1.4.2

Rewrite the expression.

$e = 0 - \frac{1}{9}$

Step 5.2.2

Subtract $\frac{1}{9}$ from $0$ .

$e = - \frac{1}{9}$

Step 6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(s - \frac{1}{3})}^{2} - \frac{1}{9}$ .

${(s - \frac{1}{3})}^{2} - \frac{1}{9}$