Algebra Examples

$x y^{2} - x + 3 y^{2} + 1 = 0$

Step 1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1

Add $x$ to both sides of the equation.

$x y^{2} + 3 y^{2} + 1 = x$

Step 1.2

Subtract $1$ from both sides of the equation.

$x y^{2} + 3 y^{2} = x - 1$

Step 2

Factor $y^{2}$ out of $x y^{2} + 3 y^{2}$ .

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Step 2.1

Factor $y^{2}$ out of $x y^{2}$ .

$y^{2} x + 3 y^{2} = x - 1$

Step 2.2

Factor $y^{2}$ out of $3 y^{2}$ .

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

Step 2.3

Factor $y^{2}$ out of $y^{2} x + y^{2} \cdot 3$ .

$y^{2} (x + 3) = x - 1$

Step 3

Divide each term in $y^{2} (x + 3) = x - 1$ by $x + 3$ and simplify.

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Step 3.1

Divide each term in $y^{2} (x + 3) = x - 1$ by $x + 3$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x + 3$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

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

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Combine the numerators over the common denominator.

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

Step 4

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

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

Step 5

Simplify $\pm \sqrt{\frac{x - 1}{x + 3}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{x - 1}{x + 3}}$ as $\frac{\sqrt{x - 1}}{\sqrt{x + 3}}$ .

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

Step 5.2

Multiply $\frac{\sqrt{x - 1}}{\sqrt{x + 3}}$ by $\frac{\sqrt{x + 3}}{\sqrt{x + 3}}$ .

$y = \pm \frac{\sqrt{x - 1}}{\sqrt{x + 3}} \cdot \frac{\sqrt{x + 3}}{\sqrt{x + 3}}$

Step 5.3

Combine and simplify the denominator.

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Step 5.3.1

Multiply $\frac{\sqrt{x - 1}}{\sqrt{x + 3}}$ by $\frac{\sqrt{x + 3}}{\sqrt{x + 3}}$ .

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.3.4

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

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

Step 5.3.5

Add $1$ and $1$ .

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

Step 5.3.6

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

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Step 5.3.6.1

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

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

Step 5.3.6.2

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

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

Step 5.3.6.3

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

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

Step 5.3.6.4

Cancel the common factor of $2$ .

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Step 5.3.6.4.1

Cancel the common factor.

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

Step 5.3.6.4.2

Rewrite the expression.

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

Step 5.3.6.5

Simplify.

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

Step 5.4

Combine using the product rule for radicals.

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

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.

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

Step 6.2

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

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

Step 6.3

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

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

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

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

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