Solve for x₂ d = square root of (x_2-x_1)^2+(y_2-y_1)^2

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

Step 1

Rewrite the equation as $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} = d$ .

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

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$ as ${({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{\frac{1}{2}}$ .

${({({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{\frac{1}{2}})}^{2} = d^{2}$

Simplify the left side.

Simplify ${({({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{\frac{1}{2} \cdot 2} = d^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{\frac{1}{2} \cdot 2} = d^{2}$

Rewrite the expression.

${({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{1} = d^{2}$

Simplify.

${(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} = d^{2}$

Step 4

Solve for $x_{2}$ .

Subtract ${(y_{2} - y_{1})}^{2}$ from both sides of the equation.

${(x_{2} - x_{1})}^{2} = d^{2} - {(y_{2} - y_{1})}^{2}$

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

$x_{2} - x_{1} = \pm \sqrt{d^{2} - {(y_{2} - y_{1})}^{2}}$

Simplify $\pm \sqrt{d^{2} - {(y_{2} - y_{1})}^{2}}$ .

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = d$ and $b = y_{2} - y_{1}$ .

$x_{2} - x_{1} = \pm \sqrt{(d + y_{2} - y_{1}) (d - (y_{2} - y_{1}))}$

Simplify.

Apply the distributive property.

$x_{2} - x_{1} = \pm \sqrt{(d + y_{2} - y_{1}) (d - y_{2} - - y_{1})}$

Multiply $- - y_{1}$ .

Multiply $- 1$ by $- 1$ .

$x_{2} - x_{1} = \pm \sqrt{(d + y_{2} - y_{1}) (d - y_{2} + 1 y_{1})}$

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

$x_{2} - x_{1} = \pm \sqrt{(d + y_{2} - y_{1}) (d - y_{2} + y_{1})}$

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

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

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

Add $x_{1}$ to both sides of the equation.

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

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

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

Add $x_{1}$ to both sides of the equation.

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

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

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

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

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

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

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

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

Solve for x₂ d = square root of (x_2-x_1)^2+(y_2-y_1)^2

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