Their approximations are
[tex]2+\sqrt{21} \approx 6.58[/tex]
[tex]2-\sqrt{21} \approx -2.58[/tex]
when rounding to 2 decimal places
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Work Shown:
Use the distance formula to get
[tex]d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\\\\5 = \sqrt{(x-2)^2 + (1-3)^2}\\\\5 = \sqrt{(x-2)^2 + (-2)^2}\\\\5 = \sqrt{(x-2)^2 + 4}\\\\5^2 = \left(\sqrt{(x-2)^2 + 4}\right)^2\\\\25 = (x-2)^2+4\\\\25 = x^2-4x+4+4\\\\25 = x^2-4x+8\\\\25-25 = x^2-4x+8-25\\\\0 = x^2-4x-17\\\\x^2-4x-17 = 0[/tex]
From here we use the quadratic formula to solve for x
[tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-(-4)\pm\sqrt{(-4)^2-4(1)(-17)}}{2(1)}\\\\x = \frac{4\pm\sqrt{84}}{2}\\\\x = \frac{4\pm\sqrt{4*21}}{2}\\\\x = \frac{4\pm\sqrt{4}*\sqrt{21}}{2}\\\\x = \frac{4\pm2\sqrt{21}}{2}\\\\x = \frac{2(2\pm\sqrt{21})}{2}\\\\x = 2\pm\sqrt{21}\\\\x = 2+\sqrt{21} \text{ or } x = 2-\sqrt{21}\\\\[/tex]
