Pre-Algebra Examples

$x = 3 y^{2} - 9$

Step 1

Rewrite the equation as $3 y^{2} - 9 = x$ .

$3 y^{2} - 9 = x$

Step 2

Add $9$ to both sides of the equation.

$3 y^{2} = x + 9$

Step 3

Divide each term in $3 y^{2} = x + 9$ by $3$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in $3 y^{2} = x + 9$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{x}{3} + \frac{9}{3}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{3 y^{2}}{3} = \frac{x}{3} + \frac{9}{3}$

Step 3.2.1.2

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

$y^{2} = \frac{x}{3} + \frac{9}{3}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Divide $9$ by $3$ .

$y^{2} = \frac{x}{3} + 3$

Step 4

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

$y = \pm \sqrt{\frac{x}{3} + 3}$

Step 5

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

Tap for more steps...

Step 5.1

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

$y = \sqrt{\frac{x}{3} + 3}$

Step 5.2

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

$y = - \sqrt{\frac{x}{3} + 3}$

Step 5.3

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

$y = \sqrt{\frac{x}{3} + 3}$

$y = - \sqrt{\frac{x}{3} + 3}$

$y = \sqrt{\frac{x}{3} + 3}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 6

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \sqrt{\frac{x}{3} + 3 \cdot \frac{3}{3}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 7

Combine $3$ and $\frac{3}{3}$ .

$y = \sqrt{\frac{x}{3} + \frac{3 \cdot 3}{3}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 8

Combine the numerators over the common denominator.

$y = \sqrt{\frac{x + 3 \cdot 3}{3}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 9

Multiply $3$ by $3$ .

$y = \sqrt{\frac{x + 9}{3}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 10

Rewrite $\sqrt{\frac{x + 9}{3}}$ as $\frac{\sqrt{x + 9}}{\sqrt{3}}$ .

$y = \frac{\sqrt{x + 9}}{\sqrt{3}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 11

Multiply $\frac{\sqrt{x + 9}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \frac{\sqrt{x + 9}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 12

Combine and simplify the denominator.

Tap for more steps...

Step 12.1

Multiply $\frac{\sqrt{x + 9}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \frac{\sqrt{x + 9} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 12.2

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

$y = \frac{\sqrt{x + 9} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 12.3

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

$y = \frac{\sqrt{x + 9} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 12.4

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

$y = \frac{\sqrt{x + 9} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 12.5

Add $1$ and $1$ .

$y = \frac{\sqrt{x + 9} \sqrt{3}}{{\sqrt{3}}^{2}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 12.6

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

Tap for more steps...

Step 12.6.1

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

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

$y = - \sqrt{\frac{x}{3} + 3}$

Step 12.6.2

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

$y = \frac{\sqrt{x + 9} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 12.6.3

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

$y = \frac{\sqrt{x + 9} \sqrt{3}}{3^{\frac{2}{2}}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 12.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 12.6.4.1

Cancel the common factor.

$y = \frac{\sqrt{x + 9} \sqrt{3}}{3^{\frac{2}{2}}}, - \sqrt{\frac{x}{3} + 3}$

Step 12.6.4.2

Rewrite the expression.

$y = \frac{\sqrt{x + 9} \sqrt{3}}{3^{1}}$

$y = - \sqrt{\frac{x}{3} + 3}$

$y = \frac{\sqrt{x + 9} \sqrt{3}}{3^{1}}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 12.6.5

Evaluate the exponent.

$y = \frac{\sqrt{x + 9} \sqrt{3}}{3}$

$y = - \sqrt{\frac{x}{3} + 3}$

$y = \frac{\sqrt{x + 9} \sqrt{3}}{3}$

$y = - \sqrt{\frac{x}{3} + 3}$

$y = \frac{\sqrt{x + 9} \sqrt{3}}{3}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 13

Combine using the product rule for radicals.

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \sqrt{\frac{x}{3} + 3}$

Step 14

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \sqrt{\frac{x}{3} + 3 \cdot \frac{3}{3}}$

Step 15

Combine $3$ and $\frac{3}{3}$ .

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \sqrt{\frac{x}{3} + \frac{3 \cdot 3}{3}}$

Step 16

Combine the numerators over the common denominator.

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \sqrt{\frac{x + 3 \cdot 3}{3}}$

Step 17

Multiply $3$ by $3$ .

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \sqrt{\frac{x + 9}{3}}$

Step 18

Rewrite $\sqrt{\frac{x + 9}{3}}$ as $\frac{\sqrt{x + 9}}{\sqrt{3}}$ .

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{x + 9}}{\sqrt{3}}$

Step 19

Multiply $\frac{\sqrt{x + 9}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - (\frac{\sqrt{x + 9}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 20

Combine and simplify the denominator.

Tap for more steps...

Step 20.1

Multiply $\frac{\sqrt{x + 9}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{x + 9} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 20.2

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

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{x + 9} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 20.3

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

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{x + 9} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 20.4

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

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{x + 9} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 20.5

Add $1$ and $1$ .

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{x + 9} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 20.6

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

Tap for more steps...

Step 20.6.1

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

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

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

Step 20.6.2

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

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{x + 9} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 20.6.3

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

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{x + 9} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 20.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 20.6.4.1

Cancel the common factor.

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}, - \frac{\sqrt{x + 9} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 20.6.4.2

Rewrite the expression.

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{x + 9} \sqrt{3}}{3^{1}}$

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{x + 9} \sqrt{3}}{3^{1}}$

Step 20.6.5

Evaluate the exponent.

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{x + 9} \sqrt{3}}{3}$

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{x + 9} \sqrt{3}}{3}$

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{x + 9} \sqrt{3}}{3}$

Step 21

Combine using the product rule for radicals.

$y = \frac{\sqrt{(x + 9) \cdot 3}}{3}$

$y = - \frac{\sqrt{(x + 9) \cdot 3}}{3}$

Step 22

Reorder factors in $y = \frac{\sqrt{(x + 9) \cdot 3}}{3}, - \frac{\sqrt{(x + 9) \cdot 3}}{3}$ .

$y = \frac{\sqrt{3 (x + 9)}}{3}$

$y = - \frac{\sqrt{3 (x + 9)}}{3}$

Step 23

The given equation $y = \frac{\sqrt{3 (x + 9)}}{3}, - \frac{\sqrt{3 (x + 9)}}{3}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

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