Solve for x 1/(x^2-3)=y+1

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

Step 1

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{2} - 3, 1, 1$

Remove parentheses.

$x^{2} - 3, 1, 1$

The LCM of one and any expression is the expression.

$x^{2} - 3$

Step 2

Multiply each term in $\frac{1}{x^{2} - 3} = y + 1$ by $x^{2} - 3$ to eliminate the fractions.

Multiply each term in $\frac{1}{x^{2} - 3} = y + 1$ by $x^{2} - 3$ .

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

Simplify the left side.

Cancel the common factor of $x^{2} - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

Simplify each term.

Apply the distributive property.

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

Move $- 3$ to the left of $y$ .

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

Multiply $x^{2} - 3$ by $1$ .

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

Step 3

Solve the equation.

Rewrite the equation as $y x^{2} - 3 y + x^{2} - 3 = 1$ .

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

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

Add $3 y$ to both sides of the equation.

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

Add $3$ to both sides of the equation.

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

Add $1$ and $3$ .

$y x^{2} + x^{2} = 3 y + 4$

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

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

$x^{2} y + x^{2} = 3 y + 4$

Multiply by $1$ .

$x^{2} y + x^{2} \cdot 1 = 3 y + 4$

Factor $x^{2}$ out of $x^{2} y + x^{2} \cdot 1$ .

$x^{2} (y + 1) = 3 y + 4$

Divide each term in $x^{2} (y + 1) = 3 y + 4$ by $y + 1$ and simplify.

Divide each term in $x^{2} (y + 1) = 3 y + 4$ by $y + 1$ .

$\frac{x^{2} (y + 1)}{y + 1} = \frac{3 y}{y + 1} + \frac{4}{y + 1}$

Simplify the left side.

Cancel the common factor of $y + 1$ .

Cancel the common factor.

$\frac{x^{2} (y + 1)}{y + 1} = \frac{3 y}{y + 1} + \frac{4}{y + 1}$

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

$x^{2} = \frac{3 y}{y + 1} + \frac{4}{y + 1}$

Simplify the right side.

Combine the numerators over the common denominator.

$x^{2} = \frac{3 y + 4}{y + 1}$

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

$x = \pm \sqrt{\frac{3 y + 4}{y + 1}}$

Simplify $\pm \sqrt{\frac{3 y + 4}{y + 1}}$ .

Rewrite $\sqrt{\frac{3 y + 4}{y + 1}}$ as $\frac{\sqrt{3 y + 4}}{\sqrt{y + 1}}$ .

$x = \pm \frac{\sqrt{3 y + 4}}{\sqrt{y + 1}}$

Multiply $\frac{\sqrt{3 y + 4}}{\sqrt{y + 1}}$ by $\frac{\sqrt{y + 1}}{\sqrt{y + 1}}$ .

$x = \pm \frac{\sqrt{3 y + 4}}{\sqrt{y + 1}} \cdot \frac{\sqrt{y + 1}}{\sqrt{y + 1}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3 y + 4}}{\sqrt{y + 1}}$ by $\frac{\sqrt{y + 1}}{\sqrt{y + 1}}$ .

$x = \pm \frac{\sqrt{3 y + 4} \sqrt{y + 1}}{\sqrt{y + 1} \sqrt{y + 1}}$

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

$x = \pm \frac{\sqrt{3 y + 4} \sqrt{y + 1}}{{\sqrt{y + 1}}^{1} \sqrt{y + 1}}$

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

$x = \pm \frac{\sqrt{3 y + 4} \sqrt{y + 1}}{{\sqrt{y + 1}}^{1} {\sqrt{y + 1}}^{1}}$

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

$x = \pm \frac{\sqrt{3 y + 4} \sqrt{y + 1}}{{\sqrt{y + 1}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{3 y + 4} \sqrt{y + 1}}{{\sqrt{y + 1}}^{2}}$

Rewrite ${\sqrt{y + 1}}^{2}$ as $y + 1$ .

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

$x = \pm \frac{\sqrt{3 y + 4} \sqrt{y + 1}}{{({(y + 1)}^{\frac{1}{2}})}^{2}}$

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

$x = \pm \frac{\sqrt{3 y + 4} \sqrt{y + 1}}{{(y + 1)}^{\frac{1}{2} \cdot 2}}$

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

$x = \pm \frac{\sqrt{3 y + 4} \sqrt{y + 1}}{{(y + 1)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{3 y + 4} \sqrt{y + 1}}{{(y + 1)}^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{3 y + 4} \sqrt{y + 1}}{{(y + 1)}^{1}}$

Simplify.

$x = \pm \frac{\sqrt{3 y + 4} \sqrt{y + 1}}{y + 1}$

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(3 y + 4) (y + 1)}}{y + 1}$

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

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

$x = \frac{\sqrt{(3 y + 4) (y + 1)}}{y + 1}$

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

$x = - \frac{\sqrt{(3 y + 4) (y + 1)}}{y + 1}$

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

$x = \frac{\sqrt{(3 y + 4) (y + 1)}}{y + 1}$

$x = - \frac{\sqrt{(3 y + 4) (y + 1)}}{y + 1}$

$x = \frac{\sqrt{(3 y + 4) (y + 1)}}{y + 1}$

$x = - \frac{\sqrt{(3 y + 4) (y + 1)}}{y + 1}$

$x = \frac{\sqrt{(3 y + 4) (y + 1)}}{y + 1}$

$x = - \frac{\sqrt{(3 y + 4) (y + 1)}}{y + 1}$

Solve for x 1/(x^2-3)=y+1

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