Basic Math Examples

$(3 m + 2 n) (6 m - 4 n) = 18 m^{2} - 8 n^{2}$

Step 1

Simplify $(3 m + 2 n) (6 m - 4 n)$ .

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Step 1.1

Rewrite.

$0 + 0 + (3 m + 2 n) (6 m - 4 n) = 18 m^{2} - 8 n^{2}$

Step 1.2

Simplify by adding zeros.

$(3 m + 2 n) (6 m - 4 n) = 18 m^{2} - 8 n^{2}$

Step 1.3

Expand $(3 m + 2 n) (6 m - 4 n)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$3 m (6 m - 4 n) + 2 n (6 m - 4 n) = 18 m^{2} - 8 n^{2}$

Step 1.3.2

Apply the distributive property.

$3 m (6 m) + 3 m (- 4 n) + 2 n (6 m - 4 n) = 18 m^{2} - 8 n^{2}$

Step 1.3.3

Apply the distributive property.

$3 m (6 m) + 3 m (- 4 n) + 2 n (6 m) + 2 n (- 4 n) = 18 m^{2} - 8 n^{2}$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Rewrite using the commutative property of multiplication.

$3 \cdot 6 m \cdot m + 3 m (- 4 n) + 2 n (6 m) + 2 n (- 4 n) = 18 m^{2} - 8 n^{2}$

Step 1.4.1.2

Multiply $m$ by $m$ by adding the exponents.

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Step 1.4.1.2.1

Move $m$ .

$3 \cdot 6 (m \cdot m) + 3 m (- 4 n) + 2 n (6 m) + 2 n (- 4 n) = 18 m^{2} - 8 n^{2}$

Step 1.4.1.2.2

Multiply $m$ by $m$ .

$3 \cdot 6 m^{2} + 3 m (- 4 n) + 2 n (6 m) + 2 n (- 4 n) = 18 m^{2} - 8 n^{2}$

Step 1.4.1.3

Multiply $3$ by $6$ .

$18 m^{2} + 3 m (- 4 n) + 2 n (6 m) + 2 n (- 4 n) = 18 m^{2} - 8 n^{2}$

Step 1.4.1.4

Rewrite using the commutative property of multiplication.

$18 m^{2} + 3 \cdot - 4 m n + 2 n (6 m) + 2 n (- 4 n) = 18 m^{2} - 8 n^{2}$

Step 1.4.1.5

Multiply $3$ by $- 4$ .

$18 m^{2} - 12 m n + 2 n (6 m) + 2 n (- 4 n) = 18 m^{2} - 8 n^{2}$

Step 1.4.1.6

Rewrite using the commutative property of multiplication.

$18 m^{2} - 12 m n + 2 \cdot 6 n m + 2 n (- 4 n) = 18 m^{2} - 8 n^{2}$

Step 1.4.1.7

Multiply $2$ by $6$ .

$18 m^{2} - 12 m n + 12 n m + 2 n (- 4 n) = 18 m^{2} - 8 n^{2}$

Step 1.4.1.8

Rewrite using the commutative property of multiplication.

$18 m^{2} - 12 m n + 12 n m + 2 \cdot - 4 n \cdot n = 18 m^{2} - 8 n^{2}$

Step 1.4.1.9

Multiply $n$ by $n$ by adding the exponents.

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Step 1.4.1.9.1

Move $n$ .

$18 m^{2} - 12 m n + 12 n m + 2 \cdot - 4 (n \cdot n) = 18 m^{2} - 8 n^{2}$

Step 1.4.1.9.2

Multiply $n$ by $n$ .

$18 m^{2} - 12 m n + 12 n m + 2 \cdot - 4 n^{2} = 18 m^{2} - 8 n^{2}$

Step 1.4.1.10

Multiply $2$ by $- 4$ .

$18 m^{2} - 12 m n + 12 n m - 8 n^{2} = 18 m^{2} - 8 n^{2}$

Step 1.4.2

Add $- 12 m n$ and $12 n m$ .

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Step 1.4.2.1

Move $n$ .

$18 m^{2} - 12 m n + 12 m n - 8 n^{2} = 18 m^{2} - 8 n^{2}$

Step 1.4.2.2

Add $- 12 m n$ and $12 m n$ .

$18 m^{2} + 0 - 8 n^{2} = 18 m^{2} - 8 n^{2}$

Step 1.4.3

Add $18 m^{2}$ and $0$ .

$18 m^{2} - 8 n^{2} = 18 m^{2} - 8 n^{2}$

Step 2

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

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Step 2.1

Subtract $18 m^{2}$ from both sides of the equation.

$18 m^{2} - 8 n^{2} - 18 m^{2} = - 8 n^{2}$

Step 2.2

Combine the opposite terms in $18 m^{2} - 8 n^{2} - 18 m^{2}$ .

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Step 2.2.1

Subtract $18 m^{2}$ from $18 m^{2}$ .

$- 8 n^{2} + 0 = - 8 n^{2}$

Step 2.2.2

Add $- 8 n^{2}$ and $0$ .

$- 8 n^{2} = - 8 n^{2}$

Step 3

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

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Step 3.1

Divide each term in $- 8 n^{2} = - 8 n^{2}$ by $- 8$ .

$\frac{- 8 n^{2}}{- 8} = \frac{- 8 n^{2}}{- 8}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 8$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 8 n^{2}}{- 8} = \frac{- 8 n^{2}}{- 8}$

Step 3.2.1.2

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

$n^{2} = \frac{- 8 n^{2}}{- 8}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $- 8$ .

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Step 3.3.1.1

Cancel the common factor.

$n^{2} = \frac{- 8 n^{2}}{- 8}$

Step 3.3.1.2

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

$n^{2} = n^{2}$

Step 4

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| n | = | n |$

Step 5

Solve for $n$ .

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Step 5.1

Rewrite the absolute value equation as four equations without absolute value bars.

$n = n$

$n = - n$

$- n = n$

$- n = - n$

Step 5.2

After simplifying, there are only two unique equations to be solved.

$n = n$

$n = - n$

Step 5.3

Solve $n = n$ for $n$ .

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Step 5.3.1

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

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Step 5.3.1.1

Subtract $n$ from both sides of the equation.

$n - n = 0$

Step 5.3.1.2

Subtract $n$ from $n$ .

$0 = 0$

Step 5.3.2

Since $0 = 0$ , the equation will always be true.

Always true

Step 5.4

Solve $n = - n$ for $n$ .

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Step 5.4.1

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

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Step 5.4.1.1

Add $n$ to both sides of the equation.

$n + n = 0$

Step 5.4.1.2

Add $n$ and $n$ .

$2 n = 0$

Step 5.4.2

Divide each term in $2 n = 0$ by $2$ and simplify.

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Step 5.4.2.1

Divide each term in $2 n = 0$ by $2$ .

$\frac{2 n}{2} = \frac{0}{2}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{2 n}{2} = \frac{0}{2}$

Step 5.4.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{0}{2}$

Step 5.4.2.3

Simplify the right side.

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Step 5.4.2.3.1

Divide $0$ by $2$ .

$n = 0$

Step 5.5

List all of the solutions.

$n = 0$

Step 6

The variable $m$ got canceled.

All real numbers

Step 7

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$