Algebra Examples

$f (x) = - (x + 3) (x + 10)$

Step 1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1

Complete the square for $- (x + 3) (x + 10)$ .

Tap for more steps...

Step 1.1.1

Simplify the expression.

Tap for more steps...

Step 1.1.1.1

Apply the distributive property.

$(- x - 1 \cdot 3) (x + 10)$

Step 1.1.1.2

Multiply $- 1$ by $3$ .

$(- x - 3) (x + 10)$

Step 1.1.1.3

Expand $(- x - 3) (x + 10)$ using the FOIL Method.

Tap for more steps...

Step 1.1.1.3.1

Apply the distributive property.

$- x (x + 10) - 3 (x + 10)$

Step 1.1.1.3.2

Apply the distributive property.

$- x \cdot x - x \cdot 10 - 3 (x + 10)$

Step 1.1.1.3.3

Apply the distributive property.

$- x \cdot x - x \cdot 10 - 3 x - 3 \cdot 10$

Step 1.1.1.4

Simplify and combine like terms.

Tap for more steps...

Step 1.1.1.4.1

Simplify each term.

Tap for more steps...

Step 1.1.1.4.1.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.1.1.4.1.1.1

Move $x$ .

$- (x \cdot x) - x \cdot 10 - 3 x - 3 \cdot 10$

Step 1.1.1.4.1.1.2

Multiply $x$ by $x$ .

$- x^{2} - x \cdot 10 - 3 x - 3 \cdot 10$

Step 1.1.1.4.1.2

Multiply $10$ by $- 1$ .

$- x^{2} - 10 x - 3 x - 3 \cdot 10$

Step 1.1.1.4.1.3

Multiply $- 3$ by $10$ .

$- x^{2} - 10 x - 3 x - 30$

Step 1.1.1.4.2

Subtract $3 x$ from $- 10 x$ .

$- x^{2} - 13 x - 30$

Step 1.1.2

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

$a = - 1$

$b = - 13$

$c = - 30$

Step 1.1.3

Consider the vertex form of a parabola.

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

Step 1.1.4

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

Tap for more steps...

Step 1.1.4.1

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

$d = \frac{- 13}{2 \cdot - 1}$

Step 1.1.4.2

Simplify the right side.

Tap for more steps...

Step 1.1.4.2.1

Multiply $2$ by $- 1$ .

$d = \frac{- 13}{- 2}$

Step 1.1.4.2.2

Dividing two negative values results in a positive value.

$d = \frac{13}{2}$

Step 1.1.5

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

Tap for more steps...

Step 1.1.5.1

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

$e = - 30 - \frac{{(- 13)}^{2}}{4 \cdot - 1}$

Step 1.1.5.2

Simplify the right side.

Tap for more steps...

Step 1.1.5.2.1

Simplify each term.

Tap for more steps...

Step 1.1.5.2.1.1

Raise $- 13$ to the power of $2$ .

$e = - 30 - \frac{169}{4 \cdot - 1}$

Step 1.1.5.2.1.2

Multiply $4$ by $- 1$ .

$e = - 30 - \frac{169}{- 4}$

Step 1.1.5.2.1.3

Move the negative in front of the fraction.

$e = - 30 - - \frac{169}{4}$

Step 1.1.5.2.1.4

Multiply $- - \frac{169}{4}$ .

Tap for more steps...

Step 1.1.5.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = - 30 + 1 (\frac{169}{4})$

Step 1.1.5.2.1.4.2

Multiply $\frac{169}{4}$ by $1$ .

$e = - 30 + \frac{169}{4}$

Step 1.1.5.2.2

To write $- 30$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = - 30 \cdot \frac{4}{4} + \frac{169}{4}$

Step 1.1.5.2.3

Combine $- 30$ and $\frac{4}{4}$ .

$e = \frac{- 30 \cdot 4}{4} + \frac{169}{4}$

Step 1.1.5.2.4

Combine the numerators over the common denominator.

$e = \frac{- 30 \cdot 4 + 169}{4}$

Step 1.1.5.2.5

Simplify the numerator.

Tap for more steps...

Step 1.1.5.2.5.1

Multiply $- 30$ by $4$ .

$e = \frac{- 120 + 169}{4}$

Step 1.1.5.2.5.2

Add $- 120$ and $169$ .

$e = \frac{49}{4}$

Step 1.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + \frac{13}{2})}^{2} + \frac{49}{4}$ .

$- {(x + \frac{13}{2})}^{2} + \frac{49}{4}$

Step 1.2

Set $y$ equal to the new right side.

$y = - {(x + \frac{13}{2})}^{2} + \frac{49}{4}$

Step 2

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

$a = - 1$

$h = - \frac{13}{2}$

$k = \frac{49}{4}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{13}{2}, \frac{49}{4})$

Step 4