Solve the equation: \$(x - 1)/(x - 2) + (x - 2)/(x - 1) = 4\$.

The given equation

\$(x - 1)/(x - 2) + (x - 2)/(x - 1) = 4\$

To solve this equation, let’s start by multiplying through by the common the denominator, which is \$(x-1)(x-2)\$

\$ (x-1)(x-1) + (x-2)(x-2) = 4(x-1)(x-2) \$

Expanding and simplifying:

\$ (x^2 - 2x + 1) + (x^2 - 4x + 4) = 4(x^2 - 3x + 2) \$

\$ 2x^2 - 6x + 5 = 4x^2 - 12x + 8 \$

\$ 2x^2 - 6x + 3 = 0\$

Now, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula

\$ x = (-b ± \sqrt(b^2 - 4ac)) / (2a) \$

In this case, a = 2, b = - 6, and c = 3. Plugging these values into the quadratic formula, we have:

\$ x = (-(-6) ± sqrt((-6)^2 - 4 * 2 * 3)) / (2 * 2) \$

\$ x = (3 ± sqrt(3)) / 2 \$

Therefore, the solutions to the original equation are \$ x = (3 + sqrt(3)) / 2 and x = (3 - sqrt(3)) / 2 \$.

This option is the correct one because it accurately reflects the roots of the quadratic equation derived from the original problem

\$ (3 + \sqrt3)/2,(3 - \sqrt3)/2\$

This option is incorrect because it only partially matches the correct set of solutions

\$ (3 + \sqrt3)/2,(-3 - \sqrt3)/2\$

It is like option B, it is incorrect because it inaccurately represents the solutions to the equation

\$ (-3 - \sqrt3)/2,(3 - \sqrt3)/2\$

This option is incorrect because neither of the provided solutions aligns with the correct answers to the given equation

\$ (-3 + \sqrt3)/2,(-3 - \sqrt3)/2\$

Option

A

Can you summarize what you’ve understood in the above steps?