Find the roots of quadratic equation 1/x - 1/x-2 = 3.

Answer: \(x = \frac{3 + \sqrt{3}}{3}\) and \(x = \frac{3 – \sqrt{3}}{3}\).

Consider the quadriatic equation

\(\frac{1}{x}-\frac{1}{x-2}=3\)

Simplify the above equation we get

\(\frac{()x – 2 – x)}{x(x-2)}=3\)

3x(x – 2) = x – 2 – x

3x2 – 6x = -2

3x2 – 6x + 2 = 0

The above equation is in the form of ax2 + bx + c = 0

Where

a = 3

b = -6

c = 2

So value of x can be written as

\(x = \frac{-b\pm \sqrt{b^{2}}-4ac}{2a}\)

Put the values of a, b and c in the above equation we get

\(x = \frac{-(-6)\pm \sqrt{(-6^{2}})-4\times 3\times 2}{2\times 3}\) \(x = \frac{6\pm \sqrt{36 -24}}{6}\) \(x = \frac{6\pm \sqrt{12}}{6}\) \(x = \frac{6\pm 2\sqrt{3}}{6}\) \(x = \frac{3\pm \sqrt{3}}{3}\)

Hence value of x is either

\(x = \frac{3 + \sqrt{3}}{3}\) or

\(x = \frac{3 – \sqrt{3}}{3}\)

∴ the roots of the quadratic equation \(\frac{1}{x}-\frac{1}{x-2}=3\) is

\(x = \frac{3 + \sqrt{3}}{3}\) and \(x = \frac{3 – \sqrt{3}}{3}\).

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