How do I find if the roots of a quadratic equation are real and equal or real and unequal using the discriminant? If I have an equation such as $x^2+3x+4=0$, how do I find out whether the roots are real, rational, and equal or real, rational, and inequal using the discriminant?

Given the equation $ax^2+bx+c=0$ the roots are: $$ x_{1,2}=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a} $$

so you can easely see that:

1) if $D=b^2-4ac < 0$ there are no real roots ( the square root of a negative number is not a real number)

2) if $D=b^2-4ac = 0$ You have a real root $x=\dfrac{-b}{2a}$ or , better, we have two coincident real roots.

3) if $D=b^2-4ac > 0$ youfind two real distict roots and, if the coefficents $a,b,c$ are rational numbers the two roots are rational only if the quare root is a rational number, and this menas that $b^2-4ac$ must be the square of a rational number.