Algebra Examples

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

Step 1

Factor each term.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

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

$y + 1, (y + 1) (y - 1), y - 1$

Step 2.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.4

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.5

The factor for $y + 1$ is $y + 1$ itself.

$(y + 1) = y + 1$

$(y + 1)$ occurs $1$ time.

Step 2.6

The factor for $y - 1$ is $y - 1$ itself.

$(y - 1) = y - 1$

$(y - 1)$ occurs $1$ time.

Step 2.7

The LCM of $y + 1, y + 1, y - 1, y - 1$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(y + 1) (y - 1)$

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Cancel the common factor of $y + 1$ .

Tap for more steps...

Step 3.2.1.1.1

Cancel the common factor.

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

Step 3.2.1.1.2

Rewrite the expression.

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

Step 3.2.1.2

Expand $(y + 2) (y - 1)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1.2.1

Apply the distributive property.

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

Step 3.2.1.2.2

Apply the distributive property.

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

Step 3.2.1.2.3

Apply the distributive property.

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

Step 3.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 3.2.1.3.1

Simplify each term.

Tap for more steps...

Step 3.2.1.3.1.1

Multiply $y$ by $y$ .

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

Step 3.2.1.3.1.2

Move $- 1$ to the left of $y$ .

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

Step 3.2.1.3.1.3

Rewrite $- 1 y$ as $- y$ .

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

Step 3.2.1.3.1.4

Multiply $2$ by $- 1$ .

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

Step 3.2.1.3.2

Add $- y$ and $2 y$ .

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

Step 3.2.1.4

Cancel the common factor of $(y + 1) (y - 1)$ .

Tap for more steps...

Step 3.2.1.4.1

Cancel the common factor.

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

Step 3.2.1.4.2

Rewrite the expression.

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

Step 3.2.2

Simplify by adding terms.

Tap for more steps...

Step 3.2.2.1

Combine the opposite terms in $y^{2} + y - 2 + 2 y^{2} - y + 3$ .

Tap for more steps...

Step 3.2.2.1.1

Subtract $y$ from $y$ .

$y^{2} - 2 + 2 y^{2} + 0 + 3 = - \frac{y - 2}{y - 1} ((y + 1) (y - 1))$

Step 3.2.2.1.2

Add $y^{2} - 2 + 2 y^{2}$ and $0$ .

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

Step 3.2.2.2

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

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

Step 3.2.2.3

Add $- 2$ and $3$ .

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

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Simplify terms.

Tap for more steps...

Step 3.3.1.1

Cancel the common factor of $y - 1$ .

Tap for more steps...

Step 3.3.1.1.1

Move the leading negative in $- \frac{y - 2}{y - 1}$ into the numerator.

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

Step 3.3.1.1.2

Factor $y - 1$ out of $(y + 1) (y - 1)$ .

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

Step 3.3.1.1.3

Cancel the common factor.

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

Step 3.3.1.1.4

Rewrite the expression.

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

Step 3.3.1.2

Apply the distributive property.

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

Step 3.3.1.3

Multiply $- 1$ by $- 2$ .

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

Step 3.3.2

Expand $(- y + 2) (y + 1)$ using the FOIL Method.

Tap for more steps...

Step 3.3.2.1

Apply the distributive property.

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

Step 3.3.2.2

Apply the distributive property.

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

Step 3.3.2.3

Apply the distributive property.

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

Step 3.3.3

Simplify and combine like terms.

Tap for more steps...

Step 3.3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.3.1.1

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 3.3.3.1.1.1

Move $y$ .

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

Step 3.3.3.1.1.2

Multiply $y$ by $y$ .

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

Step 3.3.3.1.2

Multiply $- 1$ by $1$ .

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

Step 3.3.3.1.3

Multiply $2$ by $1$ .

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

Step 3.3.3.2

Add $- y$ and $2 y$ .

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

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 4.2

Move all terms containing $y$ to the left side of the equation.

Tap for more steps...

Step 4.2.1

Subtract $3 y^{2}$ from both sides of the equation.

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

Step 4.2.2

Subtract $3 y^{2}$ from $- y^{2}$ .

$- 4 y^{2} + y + 2 = 1$

Step 4.3

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 4.3.1

Subtract $1$ from both sides of the equation.

$- 4 y^{2} + y + 2 - 1 = 0$

Step 4.3.2

Subtract $1$ from $2$ .

$- 4 y^{2} + y + 1 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.5

Substitute the values $a = - 4$ , $b = 1$ , and $c = 1$ into the quadratic formula and solve for $y$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 4 \cdot 1)}}{2 \cdot - 4}$

Step 4.6

Simplify.

Tap for more steps...

Step 4.6.1

Simplify the numerator.

Tap for more steps...

Step 4.6.1.1

One to any power is one.

$y = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 4 \cdot 1}}{2 \cdot - 4}$

Step 4.6.1.2

Multiply $- 4 \cdot - 4 \cdot 1$ .

Tap for more steps...

Step 4.6.1.2.1

Multiply $- 4$ by $- 4$ .

$y = \frac{- 1 \pm \sqrt{1 + 16 \cdot 1}}{2 \cdot - 4}$

Step 4.6.1.2.2

Multiply $16$ by $1$ .

$y = \frac{- 1 \pm \sqrt{1 + 16}}{2 \cdot - 4}$

Step 4.6.1.3

Add $1$ and $16$ .

$y = \frac{- 1 \pm \sqrt{17}}{2 \cdot - 4}$

Step 4.6.2

Multiply $2$ by $- 4$ .

$y = \frac{- 1 \pm \sqrt{17}}{- 8}$

Step 4.6.3

Simplify $\frac{- 1 \pm \sqrt{17}}{- 8}$ .

$y = \frac{1 \pm \sqrt{17}}{8}$

Step 4.7

The final answer is the combination of both solutions.

$y = \frac{1 + \sqrt{17}}{8}, \frac{1 - \sqrt{17}}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{1 + \sqrt{17}}{8}, \frac{1 - \sqrt{17}}{8}$

Decimal Form:

$y = 0.64038820 \dots, - 0.39038820 \dots$