Algebra Examples

$\frac{x^{2} - 13 x + 40}{x^{2} - 2 x - 15} \div (x^{2} - 5 x - 24)$

Step 1

Set the denominator in $\frac{x^{2} - 13 x + 40}{x^{2} - 2 x - 15}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 2 x - 15 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x^{2} - 2 x - 15$ 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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 2.1.2

Write the factored form using these integers.

$(x - 5) (x + 3) = 0$

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 $0$ .

$x - 5 = 0$

$x + 3 = 0$

Step 2.3

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.4

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

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Step 2.4.1

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

$x + 3 = 0$

Step 2.4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.5

The final solution is all the values that make $(x - 5) (x + 3) = 0$ true.

$x = 5, - 3$

Step 3

Set the denominator in $\frac{x^{2} - 13 x + 40}{x^{2} - 2 x - 15} \div (x^{2} - 5 x - 24)$ equal to $0$ to find where the expression is undefined.

$x^{2} - 5 x - 24 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Factor $x^{2} - 5 x - 24$ using the AC method.

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Step 4.1.1

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 $- 24$ and whose sum is $- 5$ .

$- 8, 3$

Step 4.1.2

Write the factored form using these integers.

$(x - 8) (x + 3) = 0$

Step 4.2

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

$x - 8 = 0$

$x + 3 = 0$

Step 4.3

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 4.3.2

Add $8$ to both sides of the equation.

$x = 8$

Step 4.4

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

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Step 4.4.1

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

$x + 3 = 0$

Step 4.4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4.5

The final solution is all the values that make $(x - 8) (x + 3) = 0$ true.

$x = 8, - 3$

Step 5

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 $0$ .

$x = - 3, x = 5, x = 8$

Step 6