Algebra Examples

$f (x) = x^{2} - 14 x + 49$ , $g (x) = x - 7$

Step 1

Find the quotient of the functions.

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Step 1.1

Replace the function designators with the actual functions in $\frac{f (x)}{g (x)}$ .

$\frac{x^{2} - 14 x + 49}{x - 7}$

Step 1.2

Simplify.

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Step 1.2.1

Factor using the perfect square rule.

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Step 1.2.1.1

Rewrite $49$ as $7^{2}$ .

$\frac{x^{2} - 14 x + 7^{2}}{x - 7}$

Step 1.2.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 x = 2 \cdot x \cdot 7$

Step 1.2.1.3

Rewrite the polynomial.

$\frac{x^{2} - 2 \cdot x \cdot 7 + 7^{2}}{x - 7}$

Step 1.2.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 7$ .

$\frac{{(x - 7)}^{2}}{x - 7}$

Step 1.2.2

Cancel the common factor of ${(x - 7)}^{2}$ and $x - 7$ .

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Step 1.2.2.1

Factor $x - 7$ out of ${(x - 7)}^{2}$ .

$\frac{(x - 7) (x - 7)}{x - 7}$

Step 1.2.2.2

Cancel the common factors.

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Step 1.2.2.2.1

Multiply by $1$ .

$\frac{(x - 7) (x - 7)}{(x - 7) \cdot 1}$

Step 1.2.2.2.2

Cancel the common factor.

$\frac{(x - 7) (x - 7)}{(x - 7) \cdot 1}$

Step 1.2.2.2.3

Rewrite the expression.

$\frac{x - 7}{1}$

Step 1.2.2.2.4

Divide $x - 7$ by $1$ .

$x - 7$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3