Finite Math Examples

$f (x) = 2 x^{2} - 2 x - 24$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

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

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

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $2 x^{2} - 2 x - 24 = 0$ .

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

Step 1.2.2

Factor the left side of the equation.

Tap for more steps...

Step 1.2.2.1

Factor $2$ out of $2 x^{2} - 2 x - 24$ .

Tap for more steps...

Step 1.2.2.1.1

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) - 2 x - 24 = 0$

Step 1.2.2.1.2

Factor $2$ out of $- 2 x$ .

$2 (x^{2}) + 2 (- x) - 24 = 0$

Step 1.2.2.1.3

Factor $2$ out of $- 24$ .

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

Step 1.2.2.1.4

Factor $2$ out of $2 (x^{2}) + 2 (- x)$ .

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

Step 1.2.2.1.5

Factor $2$ out of $2 (x^{2} - x) + 2 (- 12)$ .

$2 (x^{2} - x - 12) = 0$

Step 1.2.2.2

Factor.

Tap for more steps...

Step 1.2.2.2.1

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

Tap for more steps...

Step 1.2.2.2.1.1

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 $- 1$ .

$- 4, 3$

Step 1.2.2.2.1.2

Write the factored form using these integers.

$2 ((x - 4) (x + 3)) = 0$

Step 1.2.2.2.2

Remove unnecessary parentheses.

$2 (x - 4) (x + 3) = 0$

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 $0$ .

$x - 4 = 0$

$x + 3 = 0$

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

$x - 4 = 0$

Step 1.2.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 1.2.5

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

Tap for more steps...

Step 1.2.5.1

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

$x + 3 = 0$

Step 1.2.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.6

The final solution is all the values that make $2 (x - 4) (x + 3) = 0$ true.

$x = 4, - 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(4, 0), (- 3, 0)$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

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

$y = 2 {(0)}^{2} - 2 \cdot 0 - 24$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = 2 \cdot 0^{2} - 2 \cdot 0 - 24$

Step 2.2.2

Remove parentheses.

$y = 2 {(0)}^{2} - 2 \cdot 0 - 24$

Step 2.2.3

Simplify $2 {(0)}^{2} - 2 \cdot 0 - 24$ .

Tap for more steps...

Step 2.2.3.1

Simplify each term.

Tap for more steps...

Step 2.2.3.1.1

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

$y = 2 \cdot 0 - 2 \cdot 0 - 24$

Step 2.2.3.1.2

Multiply $2$ by $0$ .

$y = 0 - 2 \cdot 0 - 24$

Step 2.2.3.1.3

Multiply $- 2$ by $0$ .

$y = 0 + 0 - 24$

Step 2.2.3.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 - 24$

Step 2.2.3.2.2

Subtract $24$ from $0$ .

$y = - 24$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 24)$

Step 3

List the intersections.

x-intercept(s): $(4, 0), (- 3, 0)$

y-intercept(s): $(0, - 24)$

Step 4