Algebra Examples

$(c, 0)$ and $(d, 0)$

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.

$\sqrt{{(d - c)}^{2} + {(0 - 0)}^{2}}$

Step 3

Simplify.

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Step 3.1

Rewrite ${(d - c)}^{2}$ as $(d - c) (d - c)$ .

$\sqrt{(d - c) (d - c) + {(0 - 0)}^{2}}$

Step 3.2

Expand $(d - c) (d - c)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$\sqrt{d (d - c) - c (d - c) + {(0 - 0)}^{2}}$

Step 3.2.2

Apply the distributive property.

$\sqrt{d \cdot d + d (- c) - c (d - c) + {(0 - 0)}^{2}}$

Step 3.2.3

Apply the distributive property.

$\sqrt{d \cdot d + d (- c) - c d - c (- c) + {(0 - 0)}^{2}}$

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 $d$ by $d$ .

$\sqrt{d^{2} + d (- c) - c d - c (- c) + {(0 - 0)}^{2}}$

Step 3.3.1.2

Rewrite using the commutative property of multiplication.

$\sqrt{d^{2} - d c - c d - c (- c) + {(0 - 0)}^{2}}$

Step 3.3.1.3

Rewrite using the commutative property of multiplication.

$\sqrt{d^{2} - d c - c d - 1 \cdot - 1 c \cdot c + {(0 - 0)}^{2}}$

Step 3.3.1.4

Multiply $c$ by $c$ by adding the exponents.

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Step 3.3.1.4.1

Move $c$ .

$\sqrt{d^{2} - d c - c d - 1 \cdot - 1 (c \cdot c) + {(0 - 0)}^{2}}$

Step 3.3.1.4.2

Multiply $c$ by $c$ .

$\sqrt{d^{2} - d c - c d - 1 \cdot - 1 c^{2} + {(0 - 0)}^{2}}$

Step 3.3.1.5

Multiply $- 1$ by $- 1$ .

$\sqrt{d^{2} - d c - c d + 1 c^{2} + {(0 - 0)}^{2}}$

Step 3.3.1.6

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

$\sqrt{d^{2} - d c - c d + c^{2} + {(0 - 0)}^{2}}$

Step 3.3.2

Subtract $c d$ from $- d c$ .

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Step 3.3.2.1

Move $d$ .

$\sqrt{d^{2} - c d - c d + c^{2} + {(0 - 0)}^{2}}$

Step 3.3.2.2

Subtract $c d$ from $- c d$ .

$\sqrt{d^{2} - 2 c d + c^{2} + {(0 - 0)}^{2}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Subtract $0$ from $0$ .

$\sqrt{d^{2} - 2 c d + c^{2} + 0^{2}}$

Step 3.4.2

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

$\sqrt{d^{2} - 2 c d + c^{2} + 0}$

Step 3.4.3

Add $d^{2} - 2 c d + c^{2}$ and $0$ .

$\sqrt{d^{2} - 2 c d + c^{2}}$

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.

$2 c d = 2 \cdot d \cdot c$

Step 3.5.2

Rewrite the polynomial.

$\sqrt{d^{2} - 2 \cdot d \cdot c + c^{2}}$

Step 3.5.3

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

$\sqrt{{(d - c)}^{2}}$

Step 3.6

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

$d - c$