Finite Math Examples

$2800 x - 8 x^{2} - x^{3}$

Step 1

Write $2800 x - 8 x^{2} - x^{3}$ as an equation.

$y = 2800 x - 8 x^{2} - x^{3}$

Step 2

Find the x-intercepts.

Tap for more steps...

Step 2.1

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

$0 = 2800 x - 8 x^{2} - x^{3}$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Rewrite the equation as $2800 x - 8 x^{2} - x^{3} = 0$ .

$2800 x - 8 x^{2} - x^{3} = 0$

Step 2.2.2

Factor the left side of the equation.

Tap for more steps...

Step 2.2.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$2800 u - 8 u^{2} - u^{3} = 0$

Step 2.2.2.2

Factor $u$ out of $2800 u - 8 u^{2} - u^{3}$ .

Tap for more steps...

Step 2.2.2.2.1

Factor $u$ out of $2800 u$ .

$u \cdot 2800 - 8 u^{2} - u^{3} = 0$

Step 2.2.2.2.2

Factor $u$ out of $- 8 u^{2}$ .

$u \cdot 2800 + u (- 8 u) - u^{3} = 0$

Step 2.2.2.2.3

Factor $u$ out of $- u^{3}$ .

$u \cdot 2800 + u (- 8 u) + u (- u^{2}) = 0$

Step 2.2.2.2.4

Factor $u$ out of $u \cdot 2800 + u (- 8 u)$ .

$u \cdot (2800 - 8 u) + u (- u^{2}) = 0$

Step 2.2.2.2.5

Factor $u$ out of $u \cdot (2800 - 8 u) + u (- u^{2})$ .

$u (2800 - 8 u - u^{2}) = 0$

Step 2.2.2.3

Replace all occurrences of $u$ with $x$ .

$x (2800 - 8 x - x^{2}) = 0$

Step 2.2.3

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

$x = 0$

$2800 - 8 x - x^{2} = 0$

Step 2.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.5

Set $2800 - 8 x - x^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.2.5.1

Set $2800 - 8 x - x^{2}$ equal to $0$ .

$2800 - 8 x - x^{2} = 0$

Step 2.2.5.2

Solve $2800 - 8 x - x^{2} = 0$ for $x$ .

Tap for more steps...

Step 2.2.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.2.5.2.2

Substitute the values $a = - 1$ , $b = - 8$ , and $c = 2800$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (- 1 \cdot 2800)}}{2 \cdot - 1}$

Step 2.2.5.2.3

Simplify.

Tap for more steps...

Step 2.2.5.2.3.1

Simplify the numerator.

Tap for more steps...

Step 2.2.5.2.3.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 1 \cdot 2800}}{2 \cdot - 1}$

Step 2.2.5.2.3.1.2

Multiply $- 4 \cdot - 1 \cdot 2800$ .

Tap for more steps...

Step 2.2.5.2.3.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{8 \pm \sqrt{64 + 4 \cdot 2800}}{2 \cdot - 1}$

Step 2.2.5.2.3.1.2.2

Multiply $4$ by $2800$ .

$x = \frac{8 \pm \sqrt{64 + 11200}}{2 \cdot - 1}$

Step 2.2.5.2.3.1.3

Add $64$ and $11200$ .

$x = \frac{8 \pm \sqrt{11264}}{2 \cdot - 1}$

Step 2.2.5.2.3.1.4

Rewrite $11264$ as $32^{2} \cdot 11$ .

Tap for more steps...

Step 2.2.5.2.3.1.4.1

Factor $1024$ out of $11264$ .

$x = \frac{8 \pm \sqrt{1024 (11)}}{2 \cdot - 1}$

Step 2.2.5.2.3.1.4.2

Rewrite $1024$ as $32^{2}$ .

$x = \frac{8 \pm \sqrt{32^{2} \cdot 11}}{2 \cdot - 1}$

Step 2.2.5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 32 \sqrt{11}}{2 \cdot - 1}$

Step 2.2.5.2.3.2

Multiply $2$ by $- 1$ .

$x = \frac{8 \pm 32 \sqrt{11}}{- 2}$

Step 2.2.5.2.3.3

Simplify $\frac{8 \pm 32 \sqrt{11}}{- 2}$ .

$x = \frac{4 \pm 16 \sqrt{11}}{- 1}$

Step 2.2.5.2.3.4

Move the negative one from the denominator of $\frac{4 \pm 16 \sqrt{11}}{- 1}$ .

$x = - 1 \cdot (4 \pm 16 \sqrt{11})$

Step 2.2.5.2.3.5

Rewrite $- 1 \cdot (4 \pm 16 \sqrt{11})$ as $- (4 \pm 16 \sqrt{11})$ .

$x = - (4 \pm 16 \sqrt{11})$

Step 2.2.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 2.2.5.2.4.1

Simplify the numerator.

Tap for more steps...

Step 2.2.5.2.4.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 1 \cdot 2800}}{2 \cdot - 1}$

Step 2.2.5.2.4.1.2

Multiply $- 4 \cdot - 1 \cdot 2800$ .

Tap for more steps...

Step 2.2.5.2.4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{8 \pm \sqrt{64 + 4 \cdot 2800}}{2 \cdot - 1}$

Step 2.2.5.2.4.1.2.2

Multiply $4$ by $2800$ .

$x = \frac{8 \pm \sqrt{64 + 11200}}{2 \cdot - 1}$

Step 2.2.5.2.4.1.3

Add $64$ and $11200$ .

$x = \frac{8 \pm \sqrt{11264}}{2 \cdot - 1}$

Step 2.2.5.2.4.1.4

Rewrite $11264$ as $32^{2} \cdot 11$ .

Tap for more steps...

Step 2.2.5.2.4.1.4.1

Factor $1024$ out of $11264$ .

$x = \frac{8 \pm \sqrt{1024 (11)}}{2 \cdot - 1}$

Step 2.2.5.2.4.1.4.2

Rewrite $1024$ as $32^{2}$ .

$x = \frac{8 \pm \sqrt{32^{2} \cdot 11}}{2 \cdot - 1}$

Step 2.2.5.2.4.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 32 \sqrt{11}}{2 \cdot - 1}$

Step 2.2.5.2.4.2

Multiply $2$ by $- 1$ .

$x = \frac{8 \pm 32 \sqrt{11}}{- 2}$

Step 2.2.5.2.4.3

Simplify $\frac{8 \pm 32 \sqrt{11}}{- 2}$ .

$x = \frac{4 \pm 16 \sqrt{11}}{- 1}$

Step 2.2.5.2.4.4

Move the negative one from the denominator of $\frac{4 \pm 16 \sqrt{11}}{- 1}$ .

$x = - 1 \cdot (4 \pm 16 \sqrt{11})$

Step 2.2.5.2.4.5

Rewrite $- 1 \cdot (4 \pm 16 \sqrt{11})$ as $- (4 \pm 16 \sqrt{11})$ .

$x = - (4 \pm 16 \sqrt{11})$

Step 2.2.5.2.4.6

Change the $\pm$ to $+$ .

$x = - (4 + 16 \sqrt{11})$

Step 2.2.5.2.4.7

Apply the distributive property.

$x = - 1 \cdot 4 - (16 \sqrt{11})$

Step 2.2.5.2.4.8

Multiply $- 1$ by $4$ .

$x = - 4 - (16 \sqrt{11})$

Step 2.2.5.2.4.9

Multiply $16$ by $- 1$ .

$x = - 4 - 16 \sqrt{11}$

Step 2.2.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 2.2.5.2.5.1

Simplify the numerator.

Tap for more steps...

Step 2.2.5.2.5.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 1 \cdot 2800}}{2 \cdot - 1}$

Step 2.2.5.2.5.1.2

Multiply $- 4 \cdot - 1 \cdot 2800$ .

Tap for more steps...

Step 2.2.5.2.5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{8 \pm \sqrt{64 + 4 \cdot 2800}}{2 \cdot - 1}$

Step 2.2.5.2.5.1.2.2

Multiply $4$ by $2800$ .

$x = \frac{8 \pm \sqrt{64 + 11200}}{2 \cdot - 1}$

Step 2.2.5.2.5.1.3

Add $64$ and $11200$ .

$x = \frac{8 \pm \sqrt{11264}}{2 \cdot - 1}$

Step 2.2.5.2.5.1.4

Rewrite $11264$ as $32^{2} \cdot 11$ .

Tap for more steps...

Step 2.2.5.2.5.1.4.1

Factor $1024$ out of $11264$ .

$x = \frac{8 \pm \sqrt{1024 (11)}}{2 \cdot - 1}$

Step 2.2.5.2.5.1.4.2

Rewrite $1024$ as $32^{2}$ .

$x = \frac{8 \pm \sqrt{32^{2} \cdot 11}}{2 \cdot - 1}$

Step 2.2.5.2.5.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 32 \sqrt{11}}{2 \cdot - 1}$

Step 2.2.5.2.5.2

Multiply $2$ by $- 1$ .

$x = \frac{8 \pm 32 \sqrt{11}}{- 2}$

Step 2.2.5.2.5.3

Simplify $\frac{8 \pm 32 \sqrt{11}}{- 2}$ .

$x = \frac{4 \pm 16 \sqrt{11}}{- 1}$

Step 2.2.5.2.5.4

Move the negative one from the denominator of $\frac{4 \pm 16 \sqrt{11}}{- 1}$ .

$x = - 1 \cdot (4 \pm 16 \sqrt{11})$

Step 2.2.5.2.5.5

Rewrite $- 1 \cdot (4 \pm 16 \sqrt{11})$ as $- (4 \pm 16 \sqrt{11})$ .

$x = - (4 \pm 16 \sqrt{11})$

Step 2.2.5.2.5.6

Change the $\pm$ to $-$ .

$x = - (4 - 16 \sqrt{11})$

Step 2.2.5.2.5.7

Apply the distributive property.

$x = - 1 \cdot 4 - (- 16 \sqrt{11})$

Step 2.2.5.2.5.8

Multiply $- 1$ by $4$ .

$x = - 4 - (- 16 \sqrt{11})$

Step 2.2.5.2.5.9

Multiply $- 16$ by $- 1$ .

$x = - 4 + 16 \sqrt{11}$

Step 2.2.5.2.6

The final answer is the combination of both solutions.

$x = - 4 - 16 \sqrt{11}, - 4 + 16 \sqrt{11}$

Step 2.2.6

The final solution is all the values that make $x (2800 - 8 x - x^{2}) = 0$ true.

$x = 0, - 4 - 16 \sqrt{11}, - 4 + 16 \sqrt{11}$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 4 - 16 \sqrt{11}, 0), (- 4 + 16 \sqrt{11}, 0)$

Step 3

Find the y-intercepts.

Tap for more steps...

Step 3.1

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

$y = 2800 (0) - 8 {(0)}^{2} - {(0)}^{3}$

Step 3.2

Solve the equation.

Tap for more steps...

Step 3.2.1

Remove parentheses.

$y = 2800 (0) - 8 \cdot 0^{2} - {(0)}^{3}$

Step 3.2.2

Remove parentheses.

$y = 2800 (0) - 8 \cdot 0^{2} - 0^{3}$

Step 3.2.3

Remove parentheses.

$y = 2800 (0) - 8 {(0)}^{2} - {(0)}^{3}$

Step 3.2.4

Simplify $2800 (0) - 8 {(0)}^{2} - {(0)}^{3}$ .

Tap for more steps...

Step 3.2.4.1

Simplify each term.

Tap for more steps...

Step 3.2.4.1.1

Multiply $2800$ by $0$ .

$y = 0 - 8 {(0)}^{2} - {(0)}^{3}$

Step 3.2.4.1.2

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

$y = 0 - 8 \cdot 0 - {(0)}^{3}$

Step 3.2.4.1.3

Multiply $- 8$ by $0$ .

$y = 0 + 0 - {(0)}^{3}$

Step 3.2.4.1.4

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

$y = 0 + 0 - 0$

Step 3.2.4.1.5

Multiply $- 1$ by $0$ .

$y = 0 + 0 + 0$

Step 3.2.4.2

Simplify by adding numbers.

Tap for more steps...

Step 3.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 3.2.4.2.2

Add $0$ and $0$ .

$y = 0$

Step 3.3

y-intercept(s) in point form.

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

Step 4

List the intersections.

x-intercept(s): $(0, 0), (- 4 - 16 \sqrt{11}, 0), (- 4 + 16 \sqrt{11}, 0)$

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

Step 5