Algebra Examples

$(m, 0)$ and $(n, 0)$

Step 1

Use the distance formula to determine the distance between the two points.

$Distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Step 2

Substitute the actual values of the points into the distance formula.

$\sqrt{{(n - m)}^{2} + {(0 - 0)}^{2}}$

Step 3

Simplify.

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Step 3.1

Rewrite ${(n - m)}^{2}$ as $(n - m) (n - m)$ .

$\sqrt{(n - m) (n - m) + {(0 - 0)}^{2}}$

Step 3.2

Expand $(n - m) (n - m)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$\sqrt{n (n - m) - m (n - m) + {(0 - 0)}^{2}}$

Step 3.2.2

Apply the distributive property.

$\sqrt{n \cdot n + n (- m) - m (n - m) + {(0 - 0)}^{2}}$

Step 3.2.3

Apply the distributive property.

$\sqrt{n \cdot n + n (- m) - m n - m (- m) + {(0 - 0)}^{2}}$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $n$ by $n$ .

$\sqrt{n^{2} + n (- m) - m n - m (- m) + {(0 - 0)}^{2}}$

Step 3.3.1.2

Rewrite using the commutative property of multiplication.

$\sqrt{n^{2} - n m - m n - m (- m) + {(0 - 0)}^{2}}$

Step 3.3.1.3

Rewrite using the commutative property of multiplication.

$\sqrt{n^{2} - n m - m n - 1 \cdot - 1 m \cdot m + {(0 - 0)}^{2}}$

Step 3.3.1.4

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

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Step 3.3.1.4.1

Move $m$ .

$\sqrt{n^{2} - n m - m n - 1 \cdot - 1 (m \cdot m) + {(0 - 0)}^{2}}$

Step 3.3.1.4.2

Multiply $m$ by $m$ .

$\sqrt{n^{2} - n m - m n - 1 \cdot - 1 m^{2} + {(0 - 0)}^{2}}$

Step 3.3.1.5

Multiply $- 1$ by $- 1$ .

$\sqrt{n^{2} - n m - m n + 1 m^{2} + {(0 - 0)}^{2}}$

Step 3.3.1.6

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

$\sqrt{n^{2} - n m - m n + m^{2} + {(0 - 0)}^{2}}$

Step 3.3.2

Subtract $m n$ from $- n m$ .

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Step 3.3.2.1

Move $n$ .

$\sqrt{n^{2} - m n - m n + m^{2} + {(0 - 0)}^{2}}$

Step 3.3.2.2

Subtract $m n$ from $- m n$ .

$\sqrt{n^{2} - 2 m n + m^{2} + {(0 - 0)}^{2}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Subtract $0$ from $0$ .

$\sqrt{n^{2} - 2 m n + m^{2} + 0^{2}}$

Step 3.4.2

Raising $0$ to any positive power yields $0$ .

$\sqrt{n^{2} - 2 m n + m^{2} + 0}$

Step 3.4.3

Add $n^{2} - 2 m n + m^{2}$ and $0$ .

$\sqrt{n^{2} - 2 m n + m^{2}}$

Step 3.5

Factor using the perfect square rule.

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Step 3.5.1

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 m n = 2 \cdot n \cdot m$

Step 3.5.2

Rewrite the polynomial.

$\sqrt{n^{2} - 2 \cdot n \cdot m + m^{2}}$

Step 3.5.3

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = n$ and $b = m$ .

$\sqrt{{(n - m)}^{2}}$

Step 3.6

Pull terms out from under the radical, assuming positive real numbers.

$n - m$