Algebra Examples

$f (x) = (x^{2} + 2 x - 8) (x - 6)$

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 $x$ .

$0 = (x^{2} + 2 x - 8) (x - 6)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $(x^{2} + 2 x - 8) (x - 6) = 0$ .

$(x^{2} + 2 x - 8) (x - 6) = 0$

Step 1.2.2

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

$x^{2} + 2 x - 8 = 0$

$x - 6 = 0$

Step 1.2.3

Set $x^{2} + 2 x - 8$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $x^{2} + 2 x - 8$ equal to $0$ .

$x^{2} + 2 x - 8 = 0$

Step 1.2.3.2

Solve $x^{2} + 2 x - 8 = 0$ for $x$ .

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Step 1.2.3.2.1

Factor $x^{2} + 2 x - 8$ using the AC method.

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Step 1.2.3.2.1.1

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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 1.2.3.2.1.2

Write the factored form using these integers.

$(x - 2) (x + 4) = 0$

Step 1.2.3.2.2

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

$x - 2 = 0$

$x + 4 = 0$

Step 1.2.3.2.3

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 1.2.3.2.3.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.2.3.2.3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.3.2.4

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

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Step 1.2.3.2.4.1

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

$x + 4 = 0$

Step 1.2.3.2.4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.2.3.2.5

The final solution is all the values that make $(x - 2) (x + 4) = 0$ true.

$x = 2, - 4$

Step 1.2.4

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 1.2.4.2

Add $6$ to both sides of the equation.

$x = 6$

Step 1.2.5

The final solution is all the values that make $(x^{2} + 2 x - 8) (x - 6) = 0$ true.

$x = 2, - 4, 6$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2, 0), (- 4, 0), (6, 0)$

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 $y$ .

$y = ({(0)}^{2} + 2 (0) - 8) ((0) - 6)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = (0^{2} + 2 (0) - 8) ((0) - 6)$

Step 2.2.2

Remove parentheses.

$y = (0^{2} + 2 (0) - 8) (0 - 6)$

Step 2.2.3

Remove parentheses.

$y = ({(0)}^{2} + 2 (0) - 8) ((0) - 6)$

Step 2.2.4

Simplify $({(0)}^{2} + 2 (0) - 8) ((0) - 6)$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

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

$y = (0 + 2 (0) - 8) ((0) - 6)$

Step 2.2.4.1.2

Multiply $2$ by $0$ .

$y = (0 + 0 - 8) ((0) - 6)$

Step 2.2.4.2

Simplify the expression.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = (0 - 8) ((0) - 6)$

Step 2.2.4.2.2

Subtract $8$ from $0$ .

$y = - 8 ((0) - 6)$

Step 2.2.4.2.3

Subtract $6$ from $0$ .

$y = - 8 \cdot - 6$

Step 2.2.4.2.4

Multiply $- 8$ by $- 6$ .

$y = 48$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 48)$

Step 3

List the intersections.

x-intercept(s): $(2, 0), (- 4, 0), (6, 0)$

y-intercept(s): $(0, 48)$

Step 4