Calculus Examples

$- 2 x^{2} + 4000 x > 700000$

Step 1

Convert the inequality to an equation.

$- 2 x^{2} + 4000 x = 700000$

Step 2

Subtract $700000$ from both sides of the equation.

$- 2 x^{2} + 4000 x - 700000 = 0$

Step 3

Factor $- 2$ out of $- 2 x^{2} + 4000 x - 700000$ .

Tap for more steps...

Step 3.1

Factor $- 2$ out of $4000 x$ .

$- 2 x^{2} - 2 (- 2000 x) - 700000 = 0$

Step 3.2

Factor $- 2$ out of $- 700000$ .

$- 2 x^{2} - 2 (- 2000 x) - 2 \cdot 350000 = 0$

Step 3.3

Factor $- 2$ out of $- 2 (x^{2}) - 2 (- 2000 x)$ .

$- 2 (x^{2} - 2000 x) - 2 \cdot 350000 = 0$

Step 3.4

Factor $- 2$ out of $- 2 (x^{2} - 2000 x) - 2 (350000)$ .

$- 2 (x^{2} - 2000 x + 350000) = 0$

Step 4

Divide each term in $- 2 (x^{2} - 2000 x + 350000) = 0$ by $- 2$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $- 2 (x^{2} - 2000 x + 350000) = 0$ by $- 2$ .

$\frac{- 2 (x^{2} - 2000 x + 350000)}{- 2} = \frac{0}{- 2}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{- 2 (x^{2} - 2000 x + 350000)}{- 2} = \frac{0}{- 2}$

Step 4.2.1.2

Divide $x^{2} - 2000 x + 350000$ by $1$ .

$x^{2} - 2000 x + 350000 = \frac{0}{- 2}$

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Divide $0$ by $- 2$ .

$x^{2} - 2000 x + 350000 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = - 2000$ , and $c = 350000$ into the quadratic formula and solve for $x$ .

$\frac{2000 \pm \sqrt{{(- 2000)}^{2} - 4 \cdot (1 \cdot 350000)}}{2 \cdot 1}$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

Raise $- 2000$ to the power of $2$ .

$x = \frac{2000 \pm \sqrt{4000000 - 4 \cdot 1 \cdot 350000}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 350000$ .

Tap for more steps...

Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2000 \pm \sqrt{4000000 - 4 \cdot 350000}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $350000$ .

$x = \frac{2000 \pm \sqrt{4000000 - 1400000}}{2 \cdot 1}$

Step 7.1.3

Subtract $1400000$ from $4000000$ .

$x = \frac{2000 \pm \sqrt{2600000}}{2 \cdot 1}$

Step 7.1.4

Rewrite $2600000$ as $200^{2} \cdot 65$ .

Tap for more steps...

Step 7.1.4.1

Factor $40000$ out of $2600000$ .

$x = \frac{2000 \pm \sqrt{40000 (65)}}{2 \cdot 1}$

Step 7.1.4.2

Rewrite $40000$ as $200^{2}$ .

$x = \frac{2000 \pm \sqrt{200^{2} \cdot 65}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{2000 \pm 200 \sqrt{65}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{2000 \pm 200 \sqrt{65}}{2}$

Step 7.3

Simplify $\frac{2000 \pm 200 \sqrt{65}}{2}$ .

$x = 1000 \pm 100 \sqrt{65}$

Step 8

The final answer is the combination of both solutions.

$x = 1000 + 100 \sqrt{65}, 1000 - 100 \sqrt{65}$

Step 9

Use each root to create test intervals.

$x < 1000 - 100 \sqrt{65}$

$1000 - 100 \sqrt{65} < x < 1000 + 100 \sqrt{65}$

$x > 1000 + 100 \sqrt{65}$

Step 10

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 10.1

Test a value on the interval $x < 1000 - 100 \sqrt{65}$ to see if it makes the inequality true.

Tap for more steps...

Step 10.1.1

Choose a value on the interval $x < 1000 - 100 \sqrt{65}$ and see if this value makes the original inequality true.

$x = 0$

Step 10.1.2

Replace $x$ with $0$ in the original inequality.

$- 2 {(0)}^{2} + 4000 (0) > 700000$

Step 10.1.3

The left side $0$ is not greater than the right side $700000$ , which means that the given statement is false.

False

Step 10.2

Test a value on the interval $1000 - 100 \sqrt{65} < x < 1000 + 100 \sqrt{65}$ to see if it makes the inequality true.

Tap for more steps...

Step 10.2.1

Choose a value on the interval $1000 - 100 \sqrt{65} < x < 1000 + 100 \sqrt{65}$ and see if this value makes the original inequality true.

$x = 196$

Step 10.2.2

Replace $x$ with $196$ in the original inequality.

$- 2 {(196)}^{2} + 4000 (196) > 700000$

Step 10.2.3

The left side $707168$ is greater than the right side $700000$ , which means that the given statement is always true.

True

Step 10.3

Test a value on the interval $x > 1000 + 100 \sqrt{65}$ to see if it makes the inequality true.

Tap for more steps...

Step 10.3.1

Choose a value on the interval $x > 1000 + 100 \sqrt{65}$ and see if this value makes the original inequality true.

$x = 1809$

Step 10.3.2

Replace $x$ with $1809$ in the original inequality.

$- 2 {(1809)}^{2} + 4000 (1809) > 700000$

Step 10.3.3

The left side $691038$ is not greater than the right side $700000$ , which means that the given statement is false.

False

Step 10.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < 1000 - 100 \sqrt{65}$ False

$1000 - 100 \sqrt{65} < x < 1000 + 100 \sqrt{65}$ True

$x > 1000 + 100 \sqrt{65}$ False

$x < 1000 - 100 \sqrt{65}$ False

$1000 - 100 \sqrt{65} < x < 1000 + 100 \sqrt{65}$ True

$x > 1000 + 100 \sqrt{65}$ False

Step 11

The solution consists of all of the true intervals.

$1000 - 100 \sqrt{65} < x < 1000 + 100 \sqrt{65}$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$1000 - 100 \sqrt{65} < x < 1000 + 100 \sqrt{65}$

Interval Notation:

$(1000 - 100 \sqrt{65}, 1000 + 100 \sqrt{65})$

Step 13