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?
 A: For $ a x^2 + b x + c= 0 $
Real and equal if $ b^2 - 4 a c = 0 $
Real and unequal if $ b^2 - 4 a c > 0 $
Complex conjugates if $ b^2 - 4 a c < 0 $
A: Ok so using $$D=b^2-4ac$$ with respective constants, you would say that p has real roots if $$D \geq 0$$
They are imaginary if $$D < 0$$
Addressing whether they are rational / irrational, use the algebra theorem that the root of any prime number is irrational, so if D is prime, then they are irrational.
A: 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.  
