Write a rule for judging if an equation has solutions by looking at it in standard form...?

Answer 1

If you have an equation in standard form, you can't tell what the roots are like in most cases. But you can find the discriminant if your equation is in standard form, and that will tell you a lot.

If your discriminant c is:

c < 0 -- imaginary roots
c = 0 -- double root
c > 0 -- real roots

So for something like x^2 - x - 2 = 0, the discriminant is b^2 - 4ac = (-1)^2 - 4(1)(-2) = 1 + 8 = 9 > 0, so you will end up with real roots. In other words, you can factor it and get a solution for x:

x^2 - x - 2 = 0
(x - 2)(x + 1) = 0
x = 2, -1

And these are real roots.

If that had been x^2 - x + 2 = 0, your discriminant would be -7, and you wouldn't be able to factor that. You wouldn't be able to use the quadratic equation either, because you'd end up with a negative under the radical (notice that the discriminant is the same as the expression under the radical).

For something like x^2 + 4x + 4 = 0, your discriminant would be b^2 - 4ac = (4)^2 - 4(1)(4) = 16 - 16 = 0. Factoring, you'd get:

x^2 + 4x + 4 = 0
(x + 2)(x + 2) = 0
x = -2

which turns out to be a double root.