Multiple choice

Say true or false: The following equation has real roots $\cfrac{1}{2x-3}-\cfrac{1}{x-5}=1, x \neq {\cfrac{3}{2},5}$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

1/(2x-3) - 1/(x-5) = 1. (x-5 - (2x-3)) / ((2x-3)(x-5)) = 1. (-x-2) / (2x^2 - 13x + 15) = 1. -x-2 = 2x^2 - 13x + 15. 2x^2 - 12x + 17 = 0. Discriminant D = (-12)^2 - 4(2)(17) = 144 - 136 = 8. Since D > 0, the roots are real.

AI explanation

Multiplying by the common denominator (2x - 3)(x - 5) simplifies the equation to (x - 5) - (2x - 3) = (2x - 3)(x - 5). Simplifying the terms yields -x - 2 = 2x^2 - 13x + 15, which rearranges to the quadratic equation 2x^2 - 12x + 17 = 0. The discriminant is (-12)^2 - 4(2)(17) = 144 - 136 = 8, and since it is positive, the equation has real roots.

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