Algebra Examples

$f (x) = \sqrt{42 - x}$ , $g (x) = x^{2} - x$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x^{2} - x)$ by substituting in the value of $g$ into $f$ .

$f (x^{2} - x) = \sqrt{42 - (x^{2} - x)}$

Step 3

Apply the distributive property.

$f (x^{2} - x) = \sqrt{42 - x^{2} + x}$

Step 4

Reorder terms.

$f (x^{2} - x) = \sqrt{- x^{2} + x + 42}$

Step 5

Factor by grouping.

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Step 5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 42 = - 42$ and whose sum is $b = 1$ .

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Step 5.1.1

Multiply by $1$ .

$f (x^{2} - x) = \sqrt{- x^{2} + 1 x + 42}$

Step 5.1.2

Rewrite $1$ as $- 6$ plus $7$

$f (x^{2} - x) = \sqrt{- x^{2} + (- 6 + 7) x + 42}$

Step 5.1.3

Apply the distributive property.

$f (x^{2} - x) = \sqrt{- x^{2} - 6 x + 7 x + 42}$

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

$f (x^{2} - x) = \sqrt{(- x^{2} - 6 x) + 7 x + 42}$

Step 5.2.2

Factor out the greatest common factor (GCF) from each group.

$f (x^{2} - x) = \sqrt{x (- x - 6) - 7 (- x - 6)}$

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $- x - 6$ .

$f (x^{2} - x) = \sqrt{(- x - 6) (x - 7)}$