Algebra Examples

$\frac{x^{2} + 3 x - 4}{x^{2} + x - 12}$

Step 1

Set the denominator in $\frac{x^{2} + 3 x - 4}{x^{2} + x - 12}$ equal to $0$ to find where the expression is undefined.

$x^{2} + x - 12 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x^{2} + x - 12$ using the AC method.

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Step 2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 2.1.2

Write the factored form using these integers.

$(x - 3) (x + 4) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 3 = 0$

$x + 4 = 0$

Step 2.3

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.4

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 2.4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.5

The final solution is all the values that make $(x - 3) (x + 4) = 0$ true.

$x = 3, - 4$

Step 3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 4, x = 3$

Step 4