How do I solve a quadratic equation like x^2 - 2x - 35 = 0 without using a calculator?

If possible, you should try to factorise a quadratic. To do this, look a the factors of the constant, in this case -35. 35 is clearly divisible by 5, and 35 / 5 = 7, so 5 and 7 are one pair of factors. These are both prime numbers, so in fact we know that they are the only pair of factors. Because the 35 in the equation is negative, one of the two factors must be negative and the other positive (a negative times a positive is a negative). If we now look at the coefficient of x in the equation, we see that it is -2; this is what we want the sum of the two factors to be. If we make the 7 negative and leave the 5 as positive, the sum of the factors is 5 - 7 = -2, which is exactly what we want, so the quadratic can be factorised!
We now rewrite the equation as (x + 5)(x - 7) = 0. If you like, you can expand the brackets and see that this is indeed the same as the original equation, we're just writing it in a different form. Now, we know that if two things multiplied together equal 0, then one (or both) of them must equal 0 (e.g. 3 x 0 = 0). So, either x + 5 = 0, in which case x = -5, or x - 7 = 0, in which case x = 7. So the two solutions are x = -5 and x = 7. You can plug these values back into the original equation to test that they do both give 0.