Linear Algebra Examples

$\frac{x^{2} + 4 x - 21}{x^{2} + 14 x + 49}$

Step 1

Set the denominator in $\frac{x^{2} + 4 x - 21}{x^{2} + 14 x + 49}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 14 x + 49 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor using the perfect square rule.

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Step 2.1.1

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

$x^{2} + 14 x + 7^{2} = 0$

Step 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 2.1.3

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 7 + 7^{2} = 0$

Step 2.1.4

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

${(x + 7)}^{2} = 0$

Step 2.2

Set the $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 2.3

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 7) \cup (- 7, \infty)$

Set-Builder Notation:

${x | x \neq - 7}$

Step 4