Pre-Algebra Examples

$49 x^{2} = 81 y^{2} + 3969$

Step 1

Rewrite the equation as $81 y^{2} + 3969 = 49 x^{2}$ .

$81 y^{2} + 3969 = 49 x^{2}$

Step 2

Subtract $3969$ from both sides of the equation.

$81 y^{2} = 49 x^{2} - 3969$

Step 3

Divide each term in $81 y^{2} = 49 x^{2} - 3969$ by $81$ and simplify.

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Step 3.1

Divide each term in $81 y^{2} = 49 x^{2} - 3969$ by $81$ .

$\frac{81 y^{2}}{81} = \frac{49 x^{2}}{81} + \frac{- 3969}{81}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $81$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{81 y^{2}}{81} = \frac{49 x^{2}}{81} + \frac{- 3969}{81}$

Step 3.2.1.2

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

$y^{2} = \frac{49 x^{2}}{81} + \frac{- 3969}{81}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $- 3969$ by $81$ .

$y^{2} = \frac{49 x^{2}}{81} - 49$

Step 4

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

$y = \pm \sqrt{\frac{49 x^{2}}{81} - 49}$

Step 5

Simplify $\pm \sqrt{\frac{49 x^{2}}{81} - 49}$ .

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Step 5.1

Factor $49$ out of $\frac{49 x^{2}}{81} - 49$ .

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Step 5.1.1

Factor $49$ out of $\frac{49 x^{2}}{81}$ .

$y = \pm \sqrt{49 \frac{x^{2}}{81} - 49}$

Step 5.1.2

Factor $49$ out of $- 49$ .

$y = \pm \sqrt{49 \frac{x^{2}}{81} + 49 (- 1)}$

Step 5.1.3

Factor $49$ out of $49 \frac{x^{2}}{81} + 49 (- 1)$ .

$y = \pm \sqrt{49 (\frac{x^{2}}{81} - 1)}$

Step 5.2

Rewrite $81$ as $9^{2}$ .

$y = \pm \sqrt{49 (\frac{x^{2}}{9^{2}} - 1)}$

Step 5.3

Rewrite $\frac{x^{2}}{9^{2}}$ as ${(\frac{x}{9})}^{2}$ .

$y = \pm \sqrt{49 ({(\frac{x}{9})}^{2} - 1)}$

Step 5.4

Rewrite $1$ as $1^{2}$ .

$y = \pm \sqrt{49 ({(\frac{x}{9})}^{2} - 1^{2})}$

Step 5.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{x}{9}$ and $b = 1$ .

$y = \pm \sqrt{49 (\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 5.6

Rewrite $49 (\frac{x}{9} + 1) (\frac{x}{9} - 1)$ as $7^{2} ((\frac{x}{3^{2}} + 1) (\frac{x}{9} - 1))$ .

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Step 5.6.1

Rewrite $49$ as $7^{2}$ .

$y = \pm \sqrt{7^{2} (\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 5.6.2

Rewrite $9$ as $3^{2}$ .

$y = \pm \sqrt{7^{2} (\frac{x}{3^{2}} + 1) (\frac{x}{9} - 1)}$

Step 5.6.3

Add parentheses.

$y = \pm \sqrt{7^{2} ((\frac{x}{3^{2}} + 1) (\frac{x}{9} - 1))}$

Step 5.7

Pull terms out from under the radical.

$y = \pm 7 \sqrt{(\frac{x}{3^{2}} + 1) (\frac{x}{9} - 1)}$

Step 5.8

Raise $3$ to the power of $2$ .

$y = \pm 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 6

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

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Step 6.1

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

$y = 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 6.2

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

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 6.3

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

$y = 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

$y = 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 7

Write $1$ as a fraction with a common denominator.

$y = 7 \sqrt{(\frac{x}{9} + \frac{9}{9}) (\frac{x}{9} - 1)}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 8

Combine the numerators over the common denominator.

$y = 7 \sqrt{\frac{x + 9}{9} \cdot (\frac{x}{9} - 1)}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = 7 \sqrt{\frac{x + 9}{9} \cdot (\frac{x}{9} - 1 \cdot \frac{9}{9})}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 10

Combine $- 1$ and $\frac{9}{9}$ .

$y = 7 \sqrt{\frac{x + 9}{9} \cdot (\frac{x}{9} + \frac{- 1 \cdot 9}{9})}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 11

Combine the numerators over the common denominator.

$y = 7 \sqrt{\frac{x + 9}{9} \cdot \frac{x - 1 \cdot 9}{9}}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 12

Multiply $- 1$ by $9$ .

$y = 7 \sqrt{\frac{x + 9}{9} \cdot \frac{x - 9}{9}}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 13

Multiply $\frac{x + 9}{9}$ by $\frac{x - 9}{9}$ .

$y = 7 \sqrt{\frac{(x + 9) (x - 9)}{9 \cdot 9}}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 14

Multiply $9$ by $9$ .

$y = 7 \sqrt{\frac{(x + 9) (x - 9)}{81}}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 15

Rewrite $\frac{(x + 9) (x - 9)}{81}$ as ${(\frac{1}{9})}^{2} ((x + 9) (x - 9))$ .

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Step 15.1

Factor the perfect power $1^{2}$ out of $(x + 9) (x - 9)$ .

$y = 7 \sqrt{\frac{1^{2} ((x + 9) (x - 9))}{81}}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 15.2

Factor the perfect power $9^{2}$ out of $81$ .

$y = 7 \sqrt{\frac{1^{2} ((x + 9) (x - 9))}{9^{2} \cdot 1}}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 15.3

Rearrange the fraction $\frac{1^{2} ((x + 9) (x - 9))}{9^{2} \cdot 1}$ .

$y = 7 \sqrt{{(\frac{1}{9})}^{2} ((x + 9) (x - 9))}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

$y = 7 \sqrt{{(\frac{1}{9})}^{2} ((x + 9) (x - 9))}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 16

Pull terms out from under the radical.

$y = 7 (\frac{1}{9} \cdot \sqrt{(x + 9) (x - 9)})$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 17

Combine $\frac{1}{9}$ and $\sqrt{(x + 9) (x - 9)}$ .

$y = 7 (\frac{\sqrt{(x + 9) (x - 9)}}{9})$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 18

Combine $7$ and $\frac{\sqrt{(x + 9) (x - 9)}}{9}$ .

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \sqrt{(\frac{x}{9} + 1) (\frac{x}{9} - 1)}$

Step 19

Write $1$ as a fraction with a common denominator.

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \sqrt{(\frac{x}{9} + \frac{9}{9}) (\frac{x}{9} - 1)}$

Step 20

Combine the numerators over the common denominator.

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \sqrt{\frac{x + 9}{9} \cdot (\frac{x}{9} - 1)}$

Step 21

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \sqrt{\frac{x + 9}{9} \cdot (\frac{x}{9} - 1 \cdot \frac{9}{9})}$

Step 22

Combine $- 1$ and $\frac{9}{9}$ .

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \sqrt{\frac{x + 9}{9} \cdot (\frac{x}{9} + \frac{- 1 \cdot 9}{9})}$

Step 23

Combine the numerators over the common denominator.

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \sqrt{\frac{x + 9}{9} \cdot \frac{x - 1 \cdot 9}{9}}$

Step 24

Multiply $- 1$ by $9$ .

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \sqrt{\frac{x + 9}{9} \cdot \frac{x - 9}{9}}$

Step 25

Multiply $\frac{x + 9}{9}$ by $\frac{x - 9}{9}$ .

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \sqrt{\frac{(x + 9) (x - 9)}{9 \cdot 9}}$

Step 26

Multiply $9$ by $9$ .

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \sqrt{\frac{(x + 9) (x - 9)}{81}}$

Step 27

Rewrite $\frac{(x + 9) (x - 9)}{81}$ as ${(\frac{1}{9})}^{2} ((x + 9) (x - 9))$ .

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Step 27.1

Factor the perfect power $1^{2}$ out of $(x + 9) (x - 9)$ .

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \sqrt{\frac{1^{2} ((x + 9) (x - 9))}{81}}$

Step 27.2

Factor the perfect power $9^{2}$ out of $81$ .

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \sqrt{\frac{1^{2} ((x + 9) (x - 9))}{9^{2} \cdot 1}}$

Step 27.3

Rearrange the fraction $\frac{1^{2} ((x + 9) (x - 9))}{9^{2} \cdot 1}$ .

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \sqrt{{(\frac{1}{9})}^{2} ((x + 9) (x - 9))}$

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \sqrt{{(\frac{1}{9})}^{2} ((x + 9) (x - 9))}$

Step 28

Pull terms out from under the radical.

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 (\frac{1}{9} \cdot \sqrt{(x + 9) (x - 9)})$

Step 29

Combine $\frac{1}{9}$ and $\sqrt{(x + 9) (x - 9)}$ .

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - 7 \frac{\sqrt{(x + 9) (x - 9)}}{9}$

Step 30

Combine $- 7$ and $\frac{\sqrt{(x + 9) (x - 9)}}{9}$ .

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = \frac{- 7 \sqrt{(x + 9) (x - 9)}}{9}$

Step 31

Move the negative in front of the fraction.

$y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

$y = - \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$

Step 32

The given equation $y = \frac{7 \sqrt{(x + 9) (x - 9)}}{9}, - \frac{7 \sqrt{(x + 9) (x - 9)}}{9}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$