Simplify square root of (x_2-x_1)^2+(y_2-y_1)^2

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

Step 1

Rewrite ${(x_{2} - x_{1})}^{2}$ as $(x_{2} - x_{1}) (x_{2} - x_{1})$ .

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

Step 2

Expand $(x_{2} - x_{1}) (x_{2} - x_{1})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x_{2}$ by $x_{2}$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

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

Multiply $x_{1}$ by $x_{1}$ by adding the exponents.

Move $x_{1}$ .

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

Multiply $x_{1}$ by $x_{1}$ .

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

Multiply $- 1$ by $- 1$ .

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

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

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

Subtract $x_{1} x_{2}$ from $- x_{2} x_{1}$ .

Move $x_{1}$ .

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

Subtract $x_{2} x_{1}$ from $- x_{2} x_{1}$ .

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

Step 4

Rewrite ${(y_{2} - y_{1})}^{2}$ as $(y_{2} - y_{1}) (y_{2} - y_{1})$ .

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

Step 5

Expand $(y_{2} - y_{1}) (y_{2} - y_{1})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $y_{2}$ by $y_{2}$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

$\sqrt{{x_{2}}^{2} - 2 x_{2} x_{1} + {x_{1}}^{2} + {y_{2}}^{2} - y_{2} y_{1} - y_{1} y_{2} - 1 \cdot - 1 y_{1} y_{1}}$

Multiply $y_{1}$ by $y_{1}$ by adding the exponents.

Move $y_{1}$ .

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

Multiply $y_{1}$ by $y_{1}$ .

$\sqrt{{x_{2}}^{2} - 2 x_{2} x_{1} + {x_{1}}^{2} + {y_{2}}^{2} - y_{2} y_{1} - y_{1} y_{2} - 1 \cdot - 1 {y_{1}}^{2}}$

Multiply $- 1$ by $- 1$ .

$\sqrt{{x_{2}}^{2} - 2 x_{2} x_{1} + {x_{1}}^{2} + {y_{2}}^{2} - y_{2} y_{1} - y_{1} y_{2} + 1 {y_{1}}^{2}}$

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

$\sqrt{{x_{2}}^{2} - 2 x_{2} x_{1} + {x_{1}}^{2} + {y_{2}}^{2} - y_{2} y_{1} - y_{1} y_{2} + {y_{1}}^{2}}$

Subtract $y_{1} y_{2}$ from $- y_{2} y_{1}$ .

Move $y_{1}$ .

$\sqrt{{x_{2}}^{2} - 2 x_{2} x_{1} + {x_{1}}^{2} + {y_{2}}^{2} - y_{2} y_{1} - 1 y_{2} y_{1} + {y_{1}}^{2}}$

Subtract $y_{2} y_{1}$ from $- y_{2} y_{1}$ .

$\sqrt{{x_{2}}^{2} - 2 x_{2} x_{1} + {x_{1}}^{2} + {y_{2}}^{2} - 2 y_{2} y_{1} + {y_{1}}^{2}}$

Simplify square root of (x_2-x_1)^2+(y_2-y_1)^2

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