Pre-Algebra Examples

$f (x) = \sqrt{16 - x^{2}} + \sqrt{1 + x}$

Step 1

Rewrite the function as an equation.

$y = \sqrt{16 - x^{2}} + \sqrt{1 + x}$

Step 2

Simplify $\sqrt{16 - x^{2}} + \sqrt{1 + x}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite $16$ as $4^{2}$ .

$y = \sqrt{4^{2} - x^{2}} + \sqrt{1 + x}$

Step 2.1.2

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

$y = \sqrt{(4 + x) (4 - x)} + \sqrt{1 + x}$

Step 2.2

Simplify with commuting.

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Step 2.2.1

Reorder $4$ and $x$ .

$y = \sqrt{(x + 4) (4 - x)} + \sqrt{1 + x}$

Step 2.2.2

Reorder $4$ and $- x$ .

$y = \sqrt{(x + 4) (- x + 4)} + \sqrt{1 + x}$

Step 2.2.3

Reorder $1$ and $x$ .

$y = \sqrt{(x + 4) (- x + 4)} + \sqrt{x + 1}$