Finite Math Examples

$x^{2} - x y + 3 y^{2} = 5$

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 $x$ .

$x^{2} - x \cdot 0 + 3 {(0)}^{2} = 5$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $x^{2} - x \cdot 0 + 3 {(0)}^{2}$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Multiply $- x \cdot 0$ .

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Step 1.2.1.1.1.1

Multiply $0$ by $- 1$ .

$x^{2} + 0 x + 3 {(0)}^{2} = 5$

Step 1.2.1.1.1.2

Multiply $0$ by $x$ .

$x^{2} + 0 + 3 {(0)}^{2} = 5$

Step 1.2.1.1.2

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

$x^{2} + 0 + 3 \cdot 0 = 5$

Step 1.2.1.1.3

Multiply $3$ by $0$ .

$x^{2} + 0 + 0 = 5$

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 $0$ .

$x^{2} + 0 = 5$

Step 1.2.1.2.2

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

$x^{2} = 5$

Step 1.2.2

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

$x = \pm \sqrt{5}$

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.

$x = \sqrt{5}$

Step 1.2.3.2

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

$x = - \sqrt{5}$

Step 1.2.3.3

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

$x = \sqrt{5}, - \sqrt{5}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{5}, 0), (- \sqrt{5}, 0)$

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 $y$ .

${(0)}^{2} + 0 \cdot y + 3 y^{2} = 5$

Step 2.2

Solve the equation.

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Step 2.2.1

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

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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 $0$ .

$0 + 0 \cdot y + 3 y^{2} = 5$

Step 2.2.1.1.2

Multiply $0$ by $y$ .

$0 + 0 + 3 y^{2} = 5$

Step 2.2.1.2

Combine the opposite terms in $0 + 0 + 3 y^{2}$ .

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Step 2.2.1.2.1

Add $0$ and $0$ .

$0 + 3 y^{2} = 5$

Step 2.2.1.2.2

Add $0$ and $3 y^{2}$ .

$3 y^{2} = 5$

Step 2.2.2

Divide each term in $3 y^{2} = 5$ by $3$ and simplify.

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Step 2.2.2.1

Divide each term in $3 y^{2} = 5$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{5}{3}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{3 y^{2}}{3} = \frac{5}{3}$

Step 2.2.2.2.1.2

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

$y^{2} = \frac{5}{3}$

Step 2.2.3

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

$y = \pm \sqrt{\frac{5}{3}}$

Step 2.2.4

Simplify $\pm \sqrt{\frac{5}{3}}$ .

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Step 2.2.4.1

Rewrite $\sqrt{\frac{5}{3}}$ as $\frac{\sqrt{5}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{5}}{\sqrt{3}}$

Step 2.2.4.2

Multiply $\frac{\sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{5}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 2.2.4.3

Combine and simplify the denominator.

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Step 2.2.4.3.1

Multiply $\frac{\sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{5} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.2.4.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.2.4.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.2.4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.2.4.3.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.2.4.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.2.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{5} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.2.4.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm \frac{\sqrt{5} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.2.4.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{\sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.2.4.3.6.4

Cancel the common factor of $2$ .

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Step 2.2.4.3.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.2.4.3.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{5} \sqrt{3}}{3^{1}}$

Step 2.2.4.3.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{5} \sqrt{3}}{3}$

Step 2.2.4.4

Simplify the numerator.

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Step 2.2.4.4.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{5 \cdot 3}}{3}$

Step 2.2.4.4.2

Multiply $5$ by $3$ .

$y = \pm \frac{\sqrt{15}}{3}$

Step 2.2.5

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

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Step 2.2.5.1

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

$y = \frac{\sqrt{15}}{3}$

Step 2.2.5.2

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

$y = - \frac{\sqrt{15}}{3}$

Step 2.2.5.3

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

$y = \frac{\sqrt{15}}{3}, - \frac{\sqrt{15}}{3}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{\sqrt{15}}{3}), (0, - \frac{\sqrt{15}}{3})$

Step 3

List the intersections.

x-intercept(s): $(\sqrt{5}, 0), (- \sqrt{5}, 0)$

y-intercept(s): $(0, \frac{\sqrt{15}}{3}), (0, - \frac{\sqrt{15}}{3})$

Step 4