Calculus Examples

$25 x^{2} - 750 x + 10000$

Step 1

Write $25 x^{2} - 750 x + 10000$ as an equation.

$y = 25 x^{2} - 750 x + 10000$

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = 25 x^{2} - 750 x + 10000$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $25 x^{2} - 750 x + 10000 = 0$ .

$25 x^{2} - 750 x + 10000 = 0$

Step 2.2.2

Factor $25$ out of $25 x^{2} - 750 x + 10000$ .

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Step 2.2.2.1

Factor $25$ out of $25 x^{2}$ .

$25 (x^{2}) - 750 x + 10000 = 0$

Step 2.2.2.2

Factor $25$ out of $- 750 x$ .

$25 (x^{2}) + 25 (- 30 x) + 10000 = 0$

Step 2.2.2.3

Factor $25$ out of $10000$ .

$25 x^{2} + 25 (- 30 x) + 25 \cdot 400 = 0$

Step 2.2.2.4

Factor $25$ out of $25 x^{2} + 25 (- 30 x)$ .

$25 (x^{2} - 30 x) + 25 \cdot 400 = 0$

Step 2.2.2.5

Factor $25$ out of $25 (x^{2} - 30 x) + 25 \cdot 400$ .

$25 (x^{2} - 30 x + 400) = 0$

Step 2.2.3

Divide each term in $25 (x^{2} - 30 x + 400) = 0$ by $25$ and simplify.

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Step 2.2.3.1

Divide each term in $25 (x^{2} - 30 x + 400) = 0$ by $25$ .

$\frac{25 (x^{2} - 30 x + 400)}{25} = \frac{0}{25}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $25$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{25 (x^{2} - 30 x + 400)}{25} = \frac{0}{25}$

Step 2.2.3.2.1.2

Divide $x^{2} - 30 x + 400$ by $1$ .

$x^{2} - 30 x + 400 = \frac{0}{25}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Divide $0$ by $25$ .

$x^{2} - 30 x + 400 = 0$

Step 2.2.4

Use the quadratic formula to find the solutions.

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

Step 2.2.5

Substitute the values $a = 1$ , $b = - 30$ , and $c = 400$ into the quadratic formula and solve for $x$ .

$\frac{30 \pm \sqrt{{(- 30)}^{2} - 4 \cdot (1 \cdot 400)}}{2 \cdot 1}$

Step 2.2.6

Simplify.

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Step 2.2.6.1

Simplify the numerator.

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Step 2.2.6.1.1

Raise $- 30$ to the power of $2$ .

$x = \frac{30 \pm \sqrt{900 - 4 \cdot 1 \cdot 400}}{2 \cdot 1}$

Step 2.2.6.1.2

Multiply $- 4 \cdot 1 \cdot 400$ .

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Step 2.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{30 \pm \sqrt{900 - 4 \cdot 400}}{2 \cdot 1}$

Step 2.2.6.1.2.2

Multiply $- 4$ by $400$ .

$x = \frac{30 \pm \sqrt{900 - 1600}}{2 \cdot 1}$

Step 2.2.6.1.3

Subtract $1600$ from $900$ .

$x = \frac{30 \pm \sqrt{- 700}}{2 \cdot 1}$

Step 2.2.6.1.4

Rewrite $- 700$ as $- 1 (700)$ .

$x = \frac{30 \pm \sqrt{- 1 \cdot 700}}{2 \cdot 1}$

Step 2.2.6.1.5

Rewrite $\sqrt{- 1 (700)}$ as $\sqrt{- 1} \cdot \sqrt{700}$ .

$x = \frac{30 \pm \sqrt{- 1} \cdot \sqrt{700}}{2 \cdot 1}$

Step 2.2.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{30 \pm i \cdot \sqrt{700}}{2 \cdot 1}$

Step 2.2.6.1.7

Rewrite $700$ as $10^{2} \cdot 7$ .

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Step 2.2.6.1.7.1

Factor $100$ out of $700$ .

$x = \frac{30 \pm i \cdot \sqrt{100 (7)}}{2 \cdot 1}$

Step 2.2.6.1.7.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{30 \pm i \cdot \sqrt{10^{2} \cdot 7}}{2 \cdot 1}$

Step 2.2.6.1.8

Pull terms out from under the radical.

$x = \frac{30 \pm i \cdot (10 \sqrt{7})}{2 \cdot 1}$

Step 2.2.6.1.9

Move $10$ to the left of $i$ .

$x = \frac{30 \pm 10 i \sqrt{7}}{2 \cdot 1}$

Step 2.2.6.2

Multiply $2$ by $1$ .

$x = \frac{30 \pm 10 i \sqrt{7}}{2}$

Step 2.2.6.3

Simplify $\frac{30 \pm 10 i \sqrt{7}}{2}$ .

$x = 15 \pm 5 i \sqrt{7}$

Step 2.2.7

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

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Step 2.2.7.1

Simplify the numerator.

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Step 2.2.7.1.1

Raise $- 30$ to the power of $2$ .

$x = \frac{30 \pm \sqrt{900 - 4 \cdot 1 \cdot 400}}{2 \cdot 1}$

Step 2.2.7.1.2

Multiply $- 4 \cdot 1 \cdot 400$ .

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Step 2.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{30 \pm \sqrt{900 - 4 \cdot 400}}{2 \cdot 1}$

Step 2.2.7.1.2.2

Multiply $- 4$ by $400$ .

$x = \frac{30 \pm \sqrt{900 - 1600}}{2 \cdot 1}$

Step 2.2.7.1.3

Subtract $1600$ from $900$ .

$x = \frac{30 \pm \sqrt{- 700}}{2 \cdot 1}$

Step 2.2.7.1.4

Rewrite $- 700$ as $- 1 (700)$ .

$x = \frac{30 \pm \sqrt{- 1 \cdot 700}}{2 \cdot 1}$

Step 2.2.7.1.5

Rewrite $\sqrt{- 1 (700)}$ as $\sqrt{- 1} \cdot \sqrt{700}$ .

$x = \frac{30 \pm \sqrt{- 1} \cdot \sqrt{700}}{2 \cdot 1}$

Step 2.2.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{30 \pm i \cdot \sqrt{700}}{2 \cdot 1}$

Step 2.2.7.1.7

Rewrite $700$ as $10^{2} \cdot 7$ .

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Step 2.2.7.1.7.1

Factor $100$ out of $700$ .

$x = \frac{30 \pm i \cdot \sqrt{100 (7)}}{2 \cdot 1}$

Step 2.2.7.1.7.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{30 \pm i \cdot \sqrt{10^{2} \cdot 7}}{2 \cdot 1}$

Step 2.2.7.1.8

Pull terms out from under the radical.

$x = \frac{30 \pm i \cdot (10 \sqrt{7})}{2 \cdot 1}$

Step 2.2.7.1.9

Move $10$ to the left of $i$ .

$x = \frac{30 \pm 10 i \sqrt{7}}{2 \cdot 1}$

Step 2.2.7.2

Multiply $2$ by $1$ .

$x = \frac{30 \pm 10 i \sqrt{7}}{2}$

Step 2.2.7.3

Simplify $\frac{30 \pm 10 i \sqrt{7}}{2}$ .

$x = 15 \pm 5 i \sqrt{7}$

Step 2.2.7.4

Change the $\pm$ to $+$ .

$x = 15 + 5 i \sqrt{7}$

Step 2.2.8

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

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Step 2.2.8.1

Simplify the numerator.

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Step 2.2.8.1.1

Raise $- 30$ to the power of $2$ .

$x = \frac{30 \pm \sqrt{900 - 4 \cdot 1 \cdot 400}}{2 \cdot 1}$

Step 2.2.8.1.2

Multiply $- 4 \cdot 1 \cdot 400$ .

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Step 2.2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{30 \pm \sqrt{900 - 4 \cdot 400}}{2 \cdot 1}$

Step 2.2.8.1.2.2

Multiply $- 4$ by $400$ .

$x = \frac{30 \pm \sqrt{900 - 1600}}{2 \cdot 1}$

Step 2.2.8.1.3

Subtract $1600$ from $900$ .

$x = \frac{30 \pm \sqrt{- 700}}{2 \cdot 1}$

Step 2.2.8.1.4

Rewrite $- 700$ as $- 1 (700)$ .

$x = \frac{30 \pm \sqrt{- 1 \cdot 700}}{2 \cdot 1}$

Step 2.2.8.1.5

Rewrite $\sqrt{- 1 (700)}$ as $\sqrt{- 1} \cdot \sqrt{700}$ .

$x = \frac{30 \pm \sqrt{- 1} \cdot \sqrt{700}}{2 \cdot 1}$

Step 2.2.8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{30 \pm i \cdot \sqrt{700}}{2 \cdot 1}$

Step 2.2.8.1.7

Rewrite $700$ as $10^{2} \cdot 7$ .

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Step 2.2.8.1.7.1

Factor $100$ out of $700$ .

$x = \frac{30 \pm i \cdot \sqrt{100 (7)}}{2 \cdot 1}$

Step 2.2.8.1.7.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{30 \pm i \cdot \sqrt{10^{2} \cdot 7}}{2 \cdot 1}$

Step 2.2.8.1.8

Pull terms out from under the radical.

$x = \frac{30 \pm i \cdot (10 \sqrt{7})}{2 \cdot 1}$

Step 2.2.8.1.9

Move $10$ to the left of $i$ .

$x = \frac{30 \pm 10 i \sqrt{7}}{2 \cdot 1}$

Step 2.2.8.2

Multiply $2$ by $1$ .

$x = \frac{30 \pm 10 i \sqrt{7}}{2}$

Step 2.2.8.3

Simplify $\frac{30 \pm 10 i \sqrt{7}}{2}$ .

$x = 15 \pm 5 i \sqrt{7}$

Step 2.2.8.4

Change the $\pm$ to $-$ .

$x = 15 - 5 i \sqrt{7}$

Step 2.2.9

The final answer is the combination of both solutions.

$x = 15 + 5 i \sqrt{7}, 15 - 5 i \sqrt{7}$

Step 2.3

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

x-intercept(s): $None$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = 25 {(0)}^{2} - 750 \cdot 0 + 10000$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = 25 \cdot 0^{2} - 750 \cdot 0 + 10000$

Step 3.2.2

Remove parentheses.

$y = 25 {(0)}^{2} - 750 \cdot 0 + 10000$

Step 3.2.3

Simplify $25 {(0)}^{2} - 750 \cdot 0 + 10000$ .

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

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

$y = 25 \cdot 0 - 750 \cdot 0 + 10000$

Step 3.2.3.1.2

Multiply $25$ by $0$ .

$y = 0 - 750 \cdot 0 + 10000$

Step 3.2.3.1.3

Multiply $- 750$ by $0$ .

$y = 0 + 0 + 10000$

Step 3.2.3.2

Simplify by adding numbers.

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Step 3.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 10000$

Step 3.2.3.2.2

Add $0$ and $10000$ .

$y = 10000$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, 10000)$

Step 4

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 10000)$

Step 5