Algebra Examples

$f (x) = x^{2} - 2 x - 3$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = x^{2} - 2 x - 3$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{2} - 2 x - 3 = 0$ .

$x^{2} - 2 x - 3 = 0$

Step 1.2.2

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 1.2.2.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 1.2.2.2

Write the factored form using these integers.

$(x - 3) (x + 1) = 0$

Step 1.2.3

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

$x - 3 = 0$

$x + 1 = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$x - 3 = 0$

Step 1.2.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.5

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

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Step 1.2.5.1

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

$x + 1 = 0$

Step 1.2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.6

The final solution is all the values that make $(x - 3) (x + 1) = 0$ true.

$x = 3, - 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(3, 0), (- 1, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = {(0)}^{2} - 2 \cdot 0 - 3$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{2} - 2 \cdot 0 - 3$

Step 2.2.2

Remove parentheses.

$y = {(0)}^{2} - 2 \cdot 0 - 3$

Step 2.2.3

Simplify ${(0)}^{2} - 2 \cdot 0 - 3$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Raising $0$ to any positive power yields $0$ .

$y = 0 - 2 \cdot 0 - 3$

Step 2.2.3.1.2

Multiply $- 2$ by $0$ .

$y = 0 + 0 - 3$

Step 2.2.3.2

Simplify by adding and subtracting.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 - 3$

Step 2.2.3.2.2

Subtract $3$ from $0$ .

$y = - 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 3)$

Step 3

List the intersections.

x-intercept(s): $(3, 0), (- 1, 0)$

y-intercept(s): $(0, - 3)$

Step 4