The equation $r+\dfrac{5}{r-1}=1+\dfrac{1}{r-1}$ has
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no root
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one integral root
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two equal roots
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two unequal, rational roots
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infinitely many roots
The equation is r + 5/(r-1) = 1 + 1/(r-1). Subtract 1/(r-1) from both sides: r + 4/(r-1) = 1. Multiply by (r-1): r(r-1) + 4 = r-1. r^2 - r + 4 = r - 1. r^2 - 2r + 5 = 0. The discriminant is b^2 - 4ac = (-2)^2 - 4(1)(5) = 4 - 20 = -16. Since the discriminant is negative, there are no real roots.
Multiply the entire equation by (r - 1) to clear the denominator, resulting in r(r - 1) + 5 = (r - 1) + 1. Simplifying both sides gives r^2 - r + 5 = r, which further reduces to the quadratic equation r^2 - 2r + 5 = 0. The discriminant of this equation is (-2)^2 - 4(1)(5) = 4 - 20 = -16. Because the discriminant is negative, the equation has no real roots.