Calculus Examples

$3 x y - x^{2} - 4 = 0$

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 $x$ .

$3 x (0) - x^{2} - 4 = 0$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $3 x (0) - x^{2} - 4$ .

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Step 1.2.1.1

Multiply $3 x (0)$ .

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Step 1.2.1.1.1

Multiply $0$ by $3$ .

$0 x - x^{2} - 4 = 0$

Step 1.2.1.1.2

Multiply $0$ by $x$ .

$0 - x^{2} - 4 = 0$

Step 1.2.1.2

Subtract $x^{2}$ from $0$ .

$- x^{2} - 4 = 0$

Step 1.2.2

Add $4$ to both sides of the equation.

$- x^{2} = 4$

Step 1.2.3

Divide each term in $- x^{2} = 4$ by $- 1$ and simplify.

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Step 1.2.3.1

Divide each term in $- x^{2} = 4$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{4}{- 1}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{4}{- 1}$

Step 1.2.3.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{4}{- 1}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $4$ by $- 1$ .

$x^{2} = - 4$

Step 1.2.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 4}$

Step 1.2.5

Simplify $\pm \sqrt{- 4}$ .

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Step 1.2.5.1

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 1.2.5.2

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

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 1.2.5.3

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

$x = \pm i \cdot \sqrt{4}$

Step 1.2.5.4

Rewrite $4$ as $2^{2}$ .

$x = \pm i \cdot \sqrt{2^{2}}$

Step 1.2.5.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 2$

Step 1.2.5.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 1.2.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2 i$

Step 1.2.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2 i$

Step 1.2.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2 i, - 2 i$

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

$3 (0) \cdot y - {(0)}^{2} - 4 = 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $3 (0) \cdot y - {(0)}^{2} - 4$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Multiply $3$ by $0$ .

$0 \cdot y - {(0)}^{2} - 4 = 0$

Step 2.2.1.1.2

Multiply $0$ by $y$ .

$0 - {(0)}^{2} - 4 = 0$

Step 2.2.1.1.3

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

$0 - 0 - 4 = 0$

Step 2.2.1.1.4

Multiply $- 1$ by $0$ .

$0 + 0 - 4 = 0$

Step 2.2.1.2

Simplify by adding and subtracting.

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Step 2.2.1.2.1

Add $0$ and $0$ .

$0 - 4 = 0$

Step 2.2.1.2.2

Subtract $4$ from $0$ .

$- 4 = 0$

Step 2.2.2

Since $- 4 \neq 0$ , there are no solutions.

No solution

Step 2.3

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

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $None$

Step 4