Basic Math Examples

$\frac{a + b}{a - b} + \frac{a - b}{a + b} = 6$

Step 1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 1.1

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

$a - b, a + b, 1$

Step 1.2

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 1.3

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

Not prime

Step 1.4

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 1.5

The factor for $a - b$ is $a - b$ itself.

$(a - b) = a - b$

$(a - b)$ occurs $1$ time.

Step 1.6

The factor for $a + b$ is $a + b$ itself.

$(a + b) = a + b$

$(a + b)$ occurs $1$ time.

Step 1.7

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

$(a - b) (a + b)$

Step 2

Multiply each term in $\frac{a + b}{a - b} + \frac{a - b}{a + b} = 6$ by $(a - b) (a + b)$ to eliminate the fractions.

Tap for more steps...

Step 2.1

Multiply each term in $\frac{a + b}{a - b} + \frac{a - b}{a + b} = 6$ by $(a - b) (a + b)$ .

$\frac{a + b}{a - b} ((a - b) (a + b)) + \frac{a - b}{a + b} ((a - b) (a + b)) = 6 ((a - b) (a + b))$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Cancel the common factor of $a - b$ .

Tap for more steps...

Step 2.2.1.1.1

Cancel the common factor.

$\frac{a + b}{a - b} ((a - b) (a + b)) + \frac{a - b}{a + b} ((a - b) (a + b)) = 6 ((a - b) (a + b))$

Step 2.2.1.1.2

Rewrite the expression.

$(a + b) (a + b) + \frac{a - b}{a + b} ((a - b) (a + b)) = 6 ((a - b) (a + b))$

Step 2.2.1.2

Raise $a + b$ to the power of $1$ .

${(a + b)}^{1} (a + b) + \frac{a - b}{a + b} ((a - b) (a + b)) = 6 ((a - b) (a + b))$

Step 2.2.1.3

Raise $a + b$ to the power of $1$ .

${(a + b)}^{1} {(a + b)}^{1} + \frac{a - b}{a + b} ((a - b) (a + b)) = 6 ((a - b) (a + b))$

Step 2.2.1.4

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

${(a + b)}^{1 + 1} + \frac{a - b}{a + b} ((a - b) (a + b)) = 6 ((a - b) (a + b))$

Step 2.2.1.5

Add $1$ and $1$ .

${(a + b)}^{2} + \frac{a - b}{a + b} ((a - b) (a + b)) = 6 ((a - b) (a + b))$

Step 2.2.1.6

Cancel the common factor of $a + b$ .

Tap for more steps...

Step 2.2.1.6.1

Factor $a + b$ out of $(a - b) (a + b)$ .

${(a + b)}^{2} + \frac{a - b}{a + b} ((a + b) (a - b)) = 6 ((a - b) (a + b))$

Step 2.2.1.6.2

Cancel the common factor.

${(a + b)}^{2} + \frac{a - b}{a + b} ((a + b) (a - b)) = 6 ((a - b) (a + b))$

Step 2.2.1.6.3

Rewrite the expression.

${(a + b)}^{2} + (a - b) (a - b) = 6 ((a - b) (a + b))$

Step 2.2.1.7

Raise $a - b$ to the power of $1$ .

${(a + b)}^{2} + {(a - b)}^{1} (a - b) = 6 ((a - b) (a + b))$

Step 2.2.1.8

Raise $a - b$ to the power of $1$ .

${(a + b)}^{2} + {(a - b)}^{1} {(a - b)}^{1} = 6 ((a - b) (a + b))$

Step 2.2.1.9

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

${(a + b)}^{2} + {(a - b)}^{1 + 1} = 6 ((a - b) (a + b))$

Step 2.2.1.10

Add $1$ and $1$ .

${(a + b)}^{2} + {(a - b)}^{2} = 6 ((a - b) (a + b))$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Expand $(a - b) (a + b)$ using the FOIL Method.

Tap for more steps...

Step 2.3.1.1

Apply the distributive property.

${(a + b)}^{2} + {(a - b)}^{2} = 6 (a (a + b) - b (a + b))$

Step 2.3.1.2

Apply the distributive property.

${(a + b)}^{2} + {(a - b)}^{2} = 6 (a \cdot a + a b - b (a + b))$

Step 2.3.1.3

Apply the distributive property.

${(a + b)}^{2} + {(a - b)}^{2} = 6 (a \cdot a + a b - b a - b \cdot b)$

Step 2.3.2

Simplify terms.

Tap for more steps...

Step 2.3.2.1

Combine the opposite terms in $a \cdot a + a b - b a - b \cdot b$ .

Tap for more steps...

Step 2.3.2.1.1

Reorder the factors in the terms $a b$ and $- b a$ .

${(a + b)}^{2} + {(a - b)}^{2} = 6 (a \cdot a + a b - a b - b \cdot b)$

Step 2.3.2.1.2

Subtract $a b$ from $a b$ .

${(a + b)}^{2} + {(a - b)}^{2} = 6 (a \cdot a + 0 - b \cdot b)$

Step 2.3.2.1.3

Add $a \cdot a$ and $0$ .

${(a + b)}^{2} + {(a - b)}^{2} = 6 (a \cdot a - b \cdot b)$

Step 2.3.2.2

Simplify each term.

Tap for more steps...

Step 2.3.2.2.1

Multiply $a$ by $a$ .

${(a + b)}^{2} + {(a - b)}^{2} = 6 (a^{2} - b \cdot b)$

Step 2.3.2.2.2

Multiply $b$ by $b$ by adding the exponents.

Tap for more steps...

Step 2.3.2.2.2.1

Move $b$ .

${(a + b)}^{2} + {(a - b)}^{2} = 6 (a^{2} - (b \cdot b))$

Step 2.3.2.2.2.2

Multiply $b$ by $b$ .

${(a + b)}^{2} + {(a - b)}^{2} = 6 (a^{2} - b^{2})$

Step 2.3.2.3

Simplify by multiplying through.

Tap for more steps...

Step 2.3.2.3.1

Apply the distributive property.

${(a + b)}^{2} + {(a - b)}^{2} = 6 a^{2} + 6 (- b^{2})$

Step 2.3.2.3.2

Multiply $- 1$ by $6$ .

${(a + b)}^{2} + {(a - b)}^{2} = 6 a^{2} - 6 b^{2}$

Step 3

Solve the equation.

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Subtract $6 a^{2}$ from both sides of the equation.

${(a + b)}^{2} + {(a - b)}^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.2

Simplify each term.

Tap for more steps...

Step 3.1.2.1

Rewrite ${(a + b)}^{2}$ as $(a + b) (a + b)$ .

$(a + b) (a + b) + {(a - b)}^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.2

Expand $(a + b) (a + b)$ using the FOIL Method.

Tap for more steps...

Step 3.1.2.2.1

Apply the distributive property.

$a (a + b) + b (a + b) + {(a - b)}^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.2.2

Apply the distributive property.

$a \cdot a + a b + b (a + b) + {(a - b)}^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.2.3

Apply the distributive property.

$a \cdot a + a b + b a + b \cdot b + {(a - b)}^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.3

Simplify and combine like terms.

Tap for more steps...

Step 3.1.2.3.1

Simplify each term.

Tap for more steps...

Step 3.1.2.3.1.1

Multiply $a$ by $a$ .

$a^{2} + a b + b a + b \cdot b + {(a - b)}^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.3.1.2

Multiply $b$ by $b$ .

$a^{2} + a b + b a + b^{2} + {(a - b)}^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.3.2

Add $a b$ and $b a$ .

Tap for more steps...

Step 3.1.2.3.2.1

Reorder $b$ and $a$ .

$a^{2} + a b + a b + b^{2} + {(a - b)}^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.3.2.2

Add $a b$ and $a b$ .

$a^{2} + 2 a b + b^{2} + {(a - b)}^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.4

Rewrite ${(a - b)}^{2}$ as $(a - b) (a - b)$ .

$a^{2} + 2 a b + b^{2} + (a - b) (a - b) - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.5

Expand $(a - b) (a - b)$ using the FOIL Method.

Tap for more steps...

Step 3.1.2.5.1

Apply the distributive property.

$a^{2} + 2 a b + b^{2} + a (a - b) - b (a - b) - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.5.2

Apply the distributive property.

$a^{2} + 2 a b + b^{2} + a \cdot a + a (- b) - b (a - b) - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.5.3

Apply the distributive property.

$a^{2} + 2 a b + b^{2} + a \cdot a + a (- b) - b a - b (- b) - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.6

Simplify and combine like terms.

Tap for more steps...

Step 3.1.2.6.1

Simplify each term.

Tap for more steps...

Step 3.1.2.6.1.1

Multiply $a$ by $a$ .

$a^{2} + 2 a b + b^{2} + a^{2} + a (- b) - b a - b (- b) - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.6.1.2

Rewrite using the commutative property of multiplication.

$a^{2} + 2 a b + b^{2} + a^{2} - a b - b a - b (- b) - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.6.1.3

Rewrite using the commutative property of multiplication.

$a^{2} + 2 a b + b^{2} + a^{2} - a b - b a - 1 \cdot - 1 b \cdot b - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.6.1.4

Multiply $b$ by $b$ by adding the exponents.

Tap for more steps...

Step 3.1.2.6.1.4.1

Move $b$ .

$a^{2} + 2 a b + b^{2} + a^{2} - a b - b a - 1 \cdot - 1 (b \cdot b) - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.6.1.4.2

Multiply $b$ by $b$ .

$a^{2} + 2 a b + b^{2} + a^{2} - a b - b a - 1 \cdot - 1 b^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.6.1.5

Multiply $- 1$ by $- 1$ .

$a^{2} + 2 a b + b^{2} + a^{2} - a b - b a + 1 b^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.6.1.6

Multiply $b^{2}$ by $1$ .

$a^{2} + 2 a b + b^{2} + a^{2} - a b - b a + b^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.6.2

Subtract $b a$ from $- a b$ .

Tap for more steps...

Step 3.1.2.6.2.1

Move $b$ .

$a^{2} + 2 a b + b^{2} + a^{2} - a b - 1 a b + b^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.2.6.2.2

Subtract $a b$ from $- a b$ .

$a^{2} + 2 a b + b^{2} + a^{2} - 2 a b + b^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.3

Combine the opposite terms in $a^{2} + 2 a b + b^{2} + a^{2} - 2 a b + b^{2} - 6 a^{2}$ .

Tap for more steps...

Step 3.1.3.1

Subtract $2 a b$ from $2 a b$ .

$a^{2} + b^{2} + a^{2} + 0 + b^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.3.2

Add $a^{2} + b^{2} + a^{2}$ and $0$ .

$a^{2} + b^{2} + a^{2} + b^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.4

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

$2 a^{2} + b^{2} + b^{2} - 6 a^{2} = - 6 b^{2}$

Step 3.1.5

Subtract $6 a^{2}$ from $2 a^{2}$ .

$- 4 a^{2} + b^{2} + b^{2} = - 6 b^{2}$

Step 3.1.6

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

$- 4 a^{2} + 2 b^{2} = - 6 b^{2}$

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

$- 4 a^{2} = - 6 b^{2} - 2 b^{2}$

Step 3.2.2

Subtract $2 b^{2}$ from $- 6 b^{2}$ .

$- 4 a^{2} = - 8 b^{2}$

Step 3.3

Divide each term in $- 4 a^{2} = - 8 b^{2}$ by $- 4$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $- 4 a^{2} = - 8 b^{2}$ by $- 4$ .

$\frac{- 4 a^{2}}{- 4} = \frac{- 8 b^{2}}{- 4}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 4 a^{2}}{- 4} = \frac{- 8 b^{2}}{- 4}$

Step 3.3.2.1.2

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

$a^{2} = \frac{- 8 b^{2}}{- 4}$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Cancel the common factor of $- 8$ and $- 4$ .

Tap for more steps...

Step 3.3.3.1.1

Factor $- 4$ out of $- 8 b^{2}$ .

$a^{2} = \frac{- 4 (2 b^{2})}{- 4}$

Step 3.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.3.3.1.2.1

Factor $- 4$ out of $- 4$ .

$a^{2} = \frac{- 4 (2 b^{2})}{- 4 (1)}$

Step 3.3.3.1.2.2

Cancel the common factor.

$a^{2} = \frac{- 4 (2 b^{2})}{- 4 \cdot 1}$

Step 3.3.3.1.2.3

Rewrite the expression.

$a^{2} = \frac{2 b^{2}}{1}$

Step 3.3.3.1.2.4

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

$a^{2} = 2 b^{2}$

Step 3.4

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

$a = \pm \sqrt{2 b^{2}}$

Step 3.5

Simplify $\pm \sqrt{2 b^{2}}$ .

Tap for more steps...

Step 3.5.1

Reorder $2$ and $b^{2}$ .

$a = \pm \sqrt{b^{2} \cdot 2}$

Step 3.5.2

Pull terms out from under the radical.

$a = \pm b \sqrt{2}$

Step 3.6

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

Tap for more steps...

Step 3.6.1

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

$a = b \sqrt{2}$

Step 3.6.2

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

$a = - b \sqrt{2}$

Step 3.6.3

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

$a = b \sqrt{2}$

$a = - b \sqrt{2}$

$a = b \sqrt{2}$

$a = - b \sqrt{2}$

$a = b \sqrt{2}$

$a = - b \sqrt{2}$