Calculus Examples

$x^{2} + x y + y^{2} = 7$

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 $x$ .

$x^{2} + x (0) + {(0)}^{2} = 7$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $x^{2} + x (0) + {(0)}^{2}$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Multiply $x$ by $0$ .

$x^{2} + 0 + {(0)}^{2} = 7$

Step 1.2.1.1.2

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

$x^{2} + 0 + 0 = 7$

Step 1.2.1.2

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

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Step 1.2.1.2.1

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

$x^{2} + 0 = 7$

Step 1.2.1.2.2

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

$x^{2} = 7$

Step 1.2.2

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

$x = \pm \sqrt{7}$

Step 1.2.3

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

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Step 1.2.3.1

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

$x = \sqrt{7}$

Step 1.2.3.2

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

$x = - \sqrt{7}$

Step 1.2.3.3

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

$x = \sqrt{7}, - \sqrt{7}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{7}, 0), (- \sqrt{7}, 0)$

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 $y$ .

${(0)}^{2} + (0) \cdot y + y^{2} = 7$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify ${(0)}^{2} + (0) \cdot y + y^{2}$ .

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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 $0$ .

$0 + (0) \cdot y + y^{2} = 7$

Step 2.2.1.1.2

Multiply $0$ by $y$ .

$0 + 0 + y^{2} = 7$

Step 2.2.1.2

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

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Step 2.2.1.2.1

Add $0$ and $0$ .

$0 + y^{2} = 7$

Step 2.2.1.2.2

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

$y^{2} = 7$

Step 2.2.2

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

$y = \pm \sqrt{7}$

Step 2.2.3

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

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Step 2.2.3.1

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

$y = \sqrt{7}$

Step 2.2.3.2

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

$y = - \sqrt{7}$

Step 2.2.3.3

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

$y = \sqrt{7}, - \sqrt{7}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \sqrt{7}), (0, - \sqrt{7})$

Step 3

List the intersections.

x-intercept(s): $(\sqrt{7}, 0), (- \sqrt{7}, 0)$

y-intercept(s): $(0, \sqrt{7}), (0, - \sqrt{7})$

Step 4