Precalculus Examples

$7 x^{2} + 26 x y + 7 y^{2} - 24 = 0$

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 $x$ .

$7 x^{2} + 26 x (0) + 7 {(0)}^{2} - 24 = 0$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $7 x^{2} + 26 x (0) + 7 {(0)}^{2} - 24$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Multiply $26 x (0)$ .

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Step 1.2.1.1.1.1

Multiply $0$ by $26$ .

$7 x^{2} + 0 x + 7 {(0)}^{2} - 24 = 0$

Step 1.2.1.1.1.2

Multiply $0$ by $x$ .

$7 x^{2} + 0 + 7 {(0)}^{2} - 24 = 0$

Step 1.2.1.1.2

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

$7 x^{2} + 0 + 7 \cdot 0 - 24 = 0$

Step 1.2.1.1.3

Multiply $7$ by $0$ .

$7 x^{2} + 0 + 0 - 24 = 0$

Step 1.2.1.2

Combine the opposite terms in $7 x^{2} + 0 + 0 - 24$ .

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Step 1.2.1.2.1

Add $7 x^{2}$ and $0$ .

$7 x^{2} + 0 - 24 = 0$

Step 1.2.1.2.2

Add $7 x^{2}$ and $0$ .

$7 x^{2} - 24 = 0$

Step 1.2.2

Add $24$ to both sides of the equation.

$7 x^{2} = 24$

Step 1.2.3

Divide each term in $7 x^{2} = 24$ by $7$ and simplify.

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Step 1.2.3.1

Divide each term in $7 x^{2} = 24$ by $7$ .

$\frac{7 x^{2}}{7} = \frac{24}{7}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $7$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{7 x^{2}}{7} = \frac{24}{7}$

Step 1.2.3.2.1.2

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

$x^{2} = \frac{24}{7}$

Step 1.2.4

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

$x = \pm \sqrt{\frac{24}{7}}$

Step 1.2.5

Simplify $\pm \sqrt{\frac{24}{7}}$ .

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Step 1.2.5.1

Rewrite $\sqrt{\frac{24}{7}}$ as $\frac{\sqrt{24}}{\sqrt{7}}$ .

$x = \pm \frac{\sqrt{24}}{\sqrt{7}}$

Step 1.2.5.2

Simplify the numerator.

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Step 1.2.5.2.1

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 1.2.5.2.1.1

Factor $4$ out of $24$ .

$x = \pm \frac{\sqrt{4 (6)}}{\sqrt{7}}$

Step 1.2.5.2.1.2

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

$x = \pm \frac{\sqrt{2^{2} \cdot 6}}{\sqrt{7}}$

Step 1.2.5.2.2

Pull terms out from under the radical.

$x = \pm \frac{2 \sqrt{6}}{\sqrt{7}}$

Step 1.2.5.3

Multiply $\frac{2 \sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm \frac{2 \sqrt{6}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 1.2.5.4

Combine and simplify the denominator.

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Step 1.2.5.4.1

Multiply $\frac{2 \sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm \frac{2 \sqrt{6} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 1.2.5.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm \frac{2 \sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 1.2.5.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm \frac{2 \sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 1.2.5.4.4

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

$x = \pm \frac{2 \sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 1.2.5.4.5

Add $1$ and $1$ .

$x = \pm \frac{2 \sqrt{6} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 1.2.5.4.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 1.2.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \pm \frac{2 \sqrt{6} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 1.2.5.4.6.2

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

$x = \pm \frac{2 \sqrt{6} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 1.2.5.4.6.3

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

$x = \pm \frac{2 \sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 1.2.5.4.6.4

Cancel the common factor of $2$ .

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Step 1.2.5.4.6.4.1

Cancel the common factor.

$x = \pm \frac{2 \sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 1.2.5.4.6.4.2

Rewrite the expression.

$x = \pm \frac{2 \sqrt{6} \sqrt{7}}{7^{1}}$

Step 1.2.5.4.6.5

Evaluate the exponent.

$x = \pm \frac{2 \sqrt{6} \sqrt{7}}{7}$

Step 1.2.5.5

Simplify the numerator.

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Step 1.2.5.5.1

Combine using the product rule for radicals.

$x = \pm \frac{2 \sqrt{7 \cdot 6}}{7}$

Step 1.2.5.5.2

Multiply $7$ by $6$ .

$x = \pm \frac{2 \sqrt{42}}{7}$

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.

$x = \frac{2 \sqrt{42}}{7}$

Step 1.2.6.2

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

$x = - \frac{2 \sqrt{42}}{7}$

Step 1.2.6.3

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

$x = \frac{2 \sqrt{42}}{7}, - \frac{2 \sqrt{42}}{7}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{2 \sqrt{42}}{7}, 0), (- \frac{2 \sqrt{42}}{7}, 0)$

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 $y$ .

$7 {(0)}^{2} + 26 (0) \cdot y + 7 y^{2} - 24 = 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $7 {(0)}^{2} + 26 (0) \cdot y + 7 y^{2} - 24$ .

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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 $0$ .

$7 \cdot 0 + 26 (0) \cdot y + 7 y^{2} - 24 = 0$

Step 2.2.1.1.2

Multiply $7$ by $0$ .

$0 + 26 (0) \cdot y + 7 y^{2} - 24 = 0$

Step 2.2.1.1.3

Multiply $26$ by $0$ .

$0 + 0 \cdot y + 7 y^{2} - 24 = 0$

Step 2.2.1.1.4

Multiply $0$ by $y$ .

$0 + 0 + 7 y^{2} - 24 = 0$

Step 2.2.1.2

Combine the opposite terms in $0 + 0 + 7 y^{2} - 24$ .

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Step 2.2.1.2.1

Add $0$ and $0$ .

$0 + 7 y^{2} - 24 = 0$

Step 2.2.1.2.2

Add $0$ and $7 y^{2}$ .

$7 y^{2} - 24 = 0$

Step 2.2.2

Add $24$ to both sides of the equation.

$7 y^{2} = 24$

Step 2.2.3

Divide each term in $7 y^{2} = 24$ by $7$ and simplify.

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Step 2.2.3.1

Divide each term in $7 y^{2} = 24$ by $7$ .

$\frac{7 y^{2}}{7} = \frac{24}{7}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $7$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{7 y^{2}}{7} = \frac{24}{7}$

Step 2.2.3.2.1.2

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

$y^{2} = \frac{24}{7}$

Step 2.2.4

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

$y = \pm \sqrt{\frac{24}{7}}$

Step 2.2.5

Simplify $\pm \sqrt{\frac{24}{7}}$ .

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Step 2.2.5.1

Rewrite $\sqrt{\frac{24}{7}}$ as $\frac{\sqrt{24}}{\sqrt{7}}$ .

$y = \pm \frac{\sqrt{24}}{\sqrt{7}}$

Step 2.2.5.2

Simplify the numerator.

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Step 2.2.5.2.1

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 2.2.5.2.1.1

Factor $4$ out of $24$ .

$y = \pm \frac{\sqrt{4 (6)}}{\sqrt{7}}$

Step 2.2.5.2.1.2

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

$y = \pm \frac{\sqrt{2^{2} \cdot 6}}{\sqrt{7}}$

Step 2.2.5.2.2

Pull terms out from under the radical.

$y = \pm \frac{2 \sqrt{6}}{\sqrt{7}}$

Step 2.2.5.3

Multiply $\frac{2 \sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm \frac{2 \sqrt{6}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 2.2.5.4

Combine and simplify the denominator.

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Step 2.2.5.4.1

Multiply $\frac{2 \sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm \frac{2 \sqrt{6} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 2.2.5.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm \frac{2 \sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 2.2.5.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm \frac{2 \sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 2.2.5.4.4

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

$y = \pm \frac{2 \sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 2.2.5.4.5

Add $1$ and $1$ .

$y = \pm \frac{2 \sqrt{6} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 2.2.5.4.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.2.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$y = \pm \frac{2 \sqrt{6} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 2.2.5.4.6.2

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

$y = \pm \frac{2 \sqrt{6} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 2.2.5.4.6.3

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

$y = \pm \frac{2 \sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.2.5.4.6.4

Cancel the common factor of $2$ .

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Step 2.2.5.4.6.4.1

Cancel the common factor.

$y = \pm \frac{2 \sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.2.5.4.6.4.2

Rewrite the expression.

$y = \pm \frac{2 \sqrt{6} \sqrt{7}}{7^{1}}$

Step 2.2.5.4.6.5

Evaluate the exponent.

$y = \pm \frac{2 \sqrt{6} \sqrt{7}}{7}$

Step 2.2.5.5

Simplify the numerator.

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Step 2.2.5.5.1

Combine using the product rule for radicals.

$y = \pm \frac{2 \sqrt{7 \cdot 6}}{7}$

Step 2.2.5.5.2

Multiply $7$ by $6$ .

$y = \pm \frac{2 \sqrt{42}}{7}$

Step 2.2.6

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

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Step 2.2.6.1

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

$y = \frac{2 \sqrt{42}}{7}$

Step 2.2.6.2

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

$y = - \frac{2 \sqrt{42}}{7}$

Step 2.2.6.3

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

$y = \frac{2 \sqrt{42}}{7}, - \frac{2 \sqrt{42}}{7}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{2 \sqrt{42}}{7}), (0, - \frac{2 \sqrt{42}}{7})$

Step 3

List the intersections.

x-intercept(s): $(\frac{2 \sqrt{42}}{7}, 0), (- \frac{2 \sqrt{42}}{7}, 0)$

y-intercept(s): $(0, \frac{2 \sqrt{42}}{7}), (0, - \frac{2 \sqrt{42}}{7})$

Step 4