The equation that cannot be solved by factoring with integer coefficients is d. x^2 - 2x - 1 = 0. Here option D is correct.
The equation x^2 - 2x - 1 = 0 cannot be solved by factoring using integer coefficients. To determine this, let's analyze each option:
a. x^2 + x - 2 = 0 can be factored as (x + 2)(x - 1) = 0, so it can be solved by factoring.
b. x^2 - x - 2 = 0 can be factored as (x - 2)(x + 1) = 0, so it can be solved by factoring.
c. x^2 - 2x + 1 = 0 can be factored as (x - 1)(x - 1) or (x - 1)^2 = 0, so it can be solved by factoring.
Now, let's focus on option d. The equation x^2 - 2x - 1 = 0 has coefficients that involve the square root of 2, making it challenging to factor using integers. If you attempt to factor it, you'll find that the roots involve irrational numbers, specifically 1 - √2 and 1 + √2. Here option D is correct.
Complete question:
Which one of the following equations cannot be solved by factoring?
a. x^2 + x - 2 = 0
b. x^2 - x - 2 = 0
c. x^2 - 2x + 1 = 0
d. x^2 - 2x - 1 = 0