Trigonometry Examples

$f (x) = 4 - {(5 - \sqrt{4 x - 2})}^{2}$

Step 1

Rewrite $4$ as $2^{2}$ .

$f (x) = 2^{2} - {(5 - \sqrt{4 x - 2})}^{2}$

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = 5 - \sqrt{4 x - 2}$ .

$f (x) = (2 + 5 - \sqrt{4 x - 2}) (2 - (5 - \sqrt{4 x - 2}))$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Factor $2$ out of $4 x - 2$ .

Tap for more steps...

Step 3.1.1

Factor $2$ out of $4 x$ .

$f (x) = (2 + 5 - \sqrt{2 (2 x) - 2}) (2 - (5 - \sqrt{4 x - 2}))$

Step 3.1.2

Factor $2$ out of $- 2$ .

$f (x) = (2 + 5 - \sqrt{2 (2 x) + 2 (- 1)}) (2 - (5 - \sqrt{4 x - 2}))$

Step 3.1.3

Factor $2$ out of $2 (2 x) + 2 (- 1)$ .

$f (x) = (2 + 5 - \sqrt{2 (2 x - 1)}) (2 - (5 - \sqrt{4 x - 2}))$

Step 3.2

Add $2$ and $5$ .

$f (x) = (7 - \sqrt{2 (2 x - 1)}) (2 - (5 - \sqrt{4 x - 2}))$

Step 3.3

Factor $2$ out of $4 x - 2$ .

Tap for more steps...

Step 3.3.1

Factor $2$ out of $4 x$ .

$f (x) = (7 - \sqrt{2 (2 x - 1)}) (2 - (5 - \sqrt{2 (2 x) - 2}))$

Step 3.3.2

Factor $2$ out of $- 2$ .

$f (x) = (7 - \sqrt{2 (2 x - 1)}) (2 - (5 - \sqrt{2 (2 x) + 2 (- 1)}))$

Step 3.3.3

Factor $2$ out of $2 (2 x) + 2 (- 1)$ .

$f (x) = (7 - \sqrt{2 (2 x - 1)}) (2 - (5 - \sqrt{2 (2 x - 1)}))$

Step 3.4

Apply the distributive property.

$f (x) = (7 - \sqrt{2 (2 x - 1)}) (2 - 1 \cdot 5 + \sqrt{2 (2 x - 1)})$

Step 3.5

Multiply $- 1$ by $5$ .

$f (x) = (7 - \sqrt{2 (2 x - 1)}) (2 - 5 + \sqrt{2 (2 x - 1)})$

Step 3.6

Multiply $- - \sqrt{2 (2 x - 1)}$ .

Tap for more steps...

Step 3.6.1

Multiply $- 1$ by $- 1$ .

$f (x) = (7 - \sqrt{2 (2 x - 1)}) (2 - 5 + 1 \sqrt{2 (2 x - 1)})$

Step 3.6.2

Multiply $\sqrt{2 (2 x - 1)}$ by $1$ .

$f (x) = (7 - \sqrt{2 (2 x - 1)}) (2 - 5 + \sqrt{2 (2 x - 1)})$

Step 3.7

Subtract $5$ from $2$ .

$f (x) = (7 - \sqrt{2 (2 x - 1)}) (- 3 + \sqrt{2 (2 x - 1)})$