Algebra Examples

$f (x) = - 2 {(x + \frac{4}{5})}^{2} - 8$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - 2 {(x + \frac{4}{5})}^{2} - 8$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Remove parentheses.

$0 = - 2 {(x + \frac{4}{5})}^{2} - 8$

Step 1.2.2

Simplify $- 2 {(x + \frac{4}{5})}^{2} - 8$ .

Tap for more steps...

Step 1.2.2.1

Simplify each term.

Tap for more steps...

Step 1.2.2.1.1

Rewrite ${(x + \frac{4}{5})}^{2}$ as $(x + \frac{4}{5}) (x + \frac{4}{5})$ .

$0 = - 2 ((x + \frac{4}{5}) (x + \frac{4}{5})) - 8$

Step 1.2.2.1.2

Expand $(x + \frac{4}{5}) (x + \frac{4}{5})$ using the FOIL Method.

Tap for more steps...

Step 1.2.2.1.2.1

Apply the distributive property.

$0 = - 2 (x (x + \frac{4}{5}) + \frac{4}{5} (x + \frac{4}{5})) - 8$

Step 1.2.2.1.2.2

Apply the distributive property.

$0 = - 2 (x \cdot x + x \frac{4}{5} + \frac{4}{5} (x + \frac{4}{5})) - 8$

Step 1.2.2.1.2.3

Apply the distributive property.

$0 = - 2 (x \cdot x + x \frac{4}{5} + \frac{4}{5} x + \frac{4}{5} \cdot \frac{4}{5}) - 8$

Step 1.2.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.2.2.1.3.1

Simplify each term.

Tap for more steps...

Step 1.2.2.1.3.1.1

Multiply $x$ by $x$ .

$0 = - 2 (x^{2} + x \frac{4}{5} + \frac{4}{5} x + \frac{4}{5} \cdot \frac{4}{5}) - 8$

Step 1.2.2.1.3.1.2

Combine $x$ and $\frac{4}{5}$ .

$0 = - 2 (x^{2} + \frac{x \cdot 4}{5} + \frac{4}{5} x + \frac{4}{5} \cdot \frac{4}{5}) - 8$

Step 1.2.2.1.3.1.3

Move $4$ to the left of $x$ .

$0 = - 2 (x^{2} + \frac{4 \cdot x}{5} + \frac{4}{5} x + \frac{4}{5} \cdot \frac{4}{5}) - 8$

Step 1.2.2.1.3.1.4

Combine $\frac{4}{5}$ and $x$ .

$0 = - 2 (x^{2} + \frac{4 x}{5} + \frac{4 x}{5} + \frac{4}{5} \cdot \frac{4}{5}) - 8$

Step 1.2.2.1.3.1.5

Multiply $\frac{4}{5} \cdot \frac{4}{5}$ .

Tap for more steps...

Step 1.2.2.1.3.1.5.1

Multiply $\frac{4}{5}$ by $\frac{4}{5}$ .

$0 = - 2 (x^{2} + \frac{4 x}{5} + \frac{4 x}{5} + \frac{4 \cdot 4}{5 \cdot 5}) - 8$

Step 1.2.2.1.3.1.5.2

Multiply $4$ by $4$ .

$0 = - 2 (x^{2} + \frac{4 x}{5} + \frac{4 x}{5} + \frac{16}{5 \cdot 5}) - 8$

Step 1.2.2.1.3.1.5.3

Multiply $5$ by $5$ .

$0 = - 2 (x^{2} + \frac{4 x}{5} + \frac{4 x}{5} + \frac{16}{25}) - 8$

Step 1.2.2.1.3.2

Add $\frac{4 x}{5}$ and $\frac{4 x}{5}$ .

$0 = - 2 (x^{2} + 2 \frac{4 x}{5} + \frac{16}{25}) - 8$

Step 1.2.2.1.4

Multiply $2 \frac{4 x}{5}$ .

Tap for more steps...

Step 1.2.2.1.4.1

Combine $2$ and $\frac{4 x}{5}$ .

$0 = - 2 (x^{2} + \frac{2 (4 x)}{5} + \frac{16}{25}) - 8$

Step 1.2.2.1.4.2

Multiply $4$ by $2$ .

$0 = - 2 (x^{2} + \frac{8 x}{5} + \frac{16}{25}) - 8$

Step 1.2.2.1.5

Apply the distributive property.

$0 = - 2 x^{2} - 2 \frac{8 x}{5} - 2 (\frac{16}{25}) - 8$

Step 1.2.2.1.6

Simplify.

Tap for more steps...

Step 1.2.2.1.6.1

Multiply $- 2 \frac{8 x}{5}$ .

Tap for more steps...

Step 1.2.2.1.6.1.1

Combine $- 2$ and $\frac{8 x}{5}$ .

$0 = - 2 x^{2} + \frac{- 2 (8 x)}{5} - 2 (\frac{16}{25}) - 8$

Step 1.2.2.1.6.1.2

Multiply $8$ by $- 2$ .

$0 = - 2 x^{2} + \frac{- 16 x}{5} - 2 (\frac{16}{25}) - 8$

Step 1.2.2.1.6.2

Multiply $- 2 (\frac{16}{25})$ .

Tap for more steps...

Step 1.2.2.1.6.2.1

Combine $- 2$ and $\frac{16}{25}$ .

$0 = - 2 x^{2} + \frac{- 16 x}{5} + \frac{- 2 \cdot 16}{25} - 8$

Step 1.2.2.1.6.2.2

Multiply $- 2$ by $16$ .

$0 = - 2 x^{2} + \frac{- 16 x}{5} + \frac{- 32}{25} - 8$

Step 1.2.2.1.7

Simplify each term.

Tap for more steps...

Step 1.2.2.1.7.1

Move the negative in front of the fraction.

$0 = - 2 x^{2} - \frac{16 x}{5} + \frac{- 32}{25} - 8$

Step 1.2.2.1.7.2

Move the negative in front of the fraction.

$0 = - 2 x^{2} - \frac{16 x}{5} - \frac{32}{25} - 8$

Step 1.2.2.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$0 = - 2 x^{2} - \frac{16 x}{5} - \frac{32}{25} - 8 \cdot \frac{25}{25}$

Step 1.2.2.3

Combine $- 8$ and $\frac{25}{25}$ .

$0 = - 2 x^{2} - \frac{16 x}{5} - \frac{32}{25} + \frac{- 8 \cdot 25}{25}$

Step 1.2.2.4

Combine the numerators over the common denominator.

$0 = - 2 x^{2} - \frac{16 x}{5} + \frac{- 32 - 8 \cdot 25}{25}$

Step 1.2.2.5

Simplify the numerator.

Tap for more steps...

Step 1.2.2.5.1

Multiply $- 8$ by $25$ .

$0 = - 2 x^{2} - \frac{16 x}{5} + \frac{- 32 - 200}{25}$

Step 1.2.2.5.2

Subtract $200$ from $- 32$ .

$0 = - 2 x^{2} - \frac{16 x}{5} + \frac{- 232}{25}$

Step 1.2.2.6

Move the negative in front of the fraction.

$0 = - 2 x^{2} - \frac{16 x}{5} - \frac{232}{25}$

Step 1.2.3

Graph each side of the equation. The solution is the x-value of the point of intersection.

No solution

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - 2 {((0) + \frac{4}{5})}^{2} - 8$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = - 2 {(0 + \frac{4}{5})}^{2} - 8$

Step 2.2.2

Remove parentheses.

$y = - 2 {((0) + \frac{4}{5})}^{2} - 8$

Step 2.2.3

Simplify $- 2 {((0) + \frac{4}{5})}^{2} - 8$ .

Tap for more steps...

Step 2.2.3.1

Simplify each term.

Tap for more steps...

Step 2.2.3.1.1

Add $0$ and $\frac{4}{5}$ .

$y = - 2 {(\frac{4}{5})}^{2} - 8$

Step 2.2.3.1.2

Apply the product rule to $\frac{4}{5}$ .

$y = - 2 \frac{4^{2}}{5^{2}} - 8$

Step 2.2.3.1.3

Raise $4$ to the power of $2$ .

$y = - 2 \frac{16}{5^{2}} - 8$

Step 2.2.3.1.4

Raise $5$ to the power of $2$ .

$y = - 2 (\frac{16}{25}) - 8$

Step 2.2.3.1.5

Multiply $- 2 (\frac{16}{25})$ .

Tap for more steps...

Step 2.2.3.1.5.1

Combine $- 2$ and $\frac{16}{25}$ .

$y = \frac{- 2 \cdot 16}{25} - 8$

Step 2.2.3.1.5.2

Multiply $- 2$ by $16$ .

$y = \frac{- 32}{25} - 8$

Step 2.2.3.1.6

Move the negative in front of the fraction.

$y = - \frac{32}{25} - 8$

Step 2.2.3.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$y = - \frac{32}{25} - 8 \cdot \frac{25}{25}$

Step 2.2.3.3

Combine $- 8$ and $\frac{25}{25}$ .

$y = - \frac{32}{25} + \frac{- 8 \cdot 25}{25}$

Step 2.2.3.4

Combine the numerators over the common denominator.

$y = \frac{- 32 - 8 \cdot 25}{25}$

Step 2.2.3.5

Simplify the numerator.

Tap for more steps...

Step 2.2.3.5.1

Multiply $- 8$ by $25$ .

$y = \frac{- 32 - 200}{25}$

Step 2.2.3.5.2

Subtract $200$ from $- 32$ .

$y = \frac{- 232}{25}$

Step 2.2.3.6

Move the negative in front of the fraction.

$y = - \frac{232}{25}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{232}{25})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{232}{25})$

Step 4