Precalculus Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $x^{2} + {(0)}^{2} + 4 x - 4 \cdot 0$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

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

$x^{2} + 0 + 4 x - 4 \cdot 0 = 0$

Step 1.2.1.1.2

Multiply $- 4$ by $0$ .

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

Step 1.2.1.2

Combine the opposite terms in $x^{2} + 0 + 4 x + 0$ .

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Step 1.2.1.2.1

Add $x^{2}$ and $0$ .

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

Step 1.2.1.2.2

Add $x^{2} + 4 x$ and $0$ .

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

Step 1.2.2

Factor $x$ out of $x^{2} + 4 x$ .

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Step 1.2.2.1

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

$x \cdot x + 4 x = 0$

Step 1.2.2.2

Factor $x$ out of $4 x$ .

$x \cdot x + x \cdot 4 = 0$

Step 1.2.2.3

Factor $x$ out of $x \cdot x + x \cdot 4$ .

$x (x + 4) = 0$

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 $0$ .

$x = 0$

$x + 4 = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

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

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Step 1.2.5.1

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

$x + 4 = 0$

Step 1.2.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.2.6

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

$x = 0, - 4$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify ${(0)}^{2} + y^{2} + 4 (0) - 4 y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Multiply $4$ by $0$ .

$0 + y^{2} + 0 - 4 y = 0$

Step 2.2.1.2

Combine the opposite terms in $0 + y^{2} + 0 - 4 y$ .

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Step 2.2.1.2.1

Add $0$ and $y^{2}$ .

$y^{2} + 0 - 4 y = 0$

Step 2.2.1.2.2

Add $y^{2}$ and $0$ .

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

Step 2.2.2

Factor $y$ out of $y^{2} - 4 y$ .

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Step 2.2.2.1

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

$y \cdot y - 4 y = 0$

Step 2.2.2.2

Factor $y$ out of $- 4 y$ .

$y \cdot y + y \cdot - 4 = 0$

Step 2.2.2.3

Factor $y$ out of $y \cdot y + y \cdot - 4$ .

$y (y - 4) = 0$

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 $0$ .

$y = 0$

$y - 4 = 0$

Step 2.2.4

Set $y$ equal to $0$ .

$y = 0$

Step 2.2.5

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

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Step 2.2.5.1

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

$y - 4 = 0$

Step 2.2.5.2

Add $4$ to both sides of the equation.

$y = 4$

Step 2.2.6

The final solution is all the values that make $y (y - 4) = 0$ true.

$y = 0, 4$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0), (0, 4)$

Step 3

List the intersections.

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

y-intercept(s): $(0, 0), (0, 4)$

Step 4