Algebra Examples

$x^{2} - 4 x - 5 = 0$

Step 1

Factor $x^{2} - 4 x - 5$ using the AC method.

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Step 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 1.2

Write the factored form using these integers.

$(x - 5) (x + 1) = 0$

Step 2

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

$x - 5 = 0$

$x + 1 = 0$

Step 3

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

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Step 3.1

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

$x - 5 = 0$

Step 3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

The final solution is all the values that make $(x - 5) (x + 1) = 0$ true.

$x = 5, - 1$

Step 6

The domain is the set of all valid $x$ values.

${x | x = 5, - 1}$

Step 7

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 8

Determine the domain and range.

Domain: ${x | x = 5, - 1}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 9