why b^2 – 4ac is called …

The number of roots of a polynomial equation is equal to its degree. Hence, a quadratic equation has 2 roots. Let α and β be the roots of the general form of the quadratic equation :ax2 + bx + c = 0. We can write:

α = (-b-√b²-4ac)/2a                 and                     β = (-b+√b²-4ac)/2a

Here a, b, and c are real and rational. Hence, the nature of the roots α and β of equation ax² + bx + c = 0 depends on the quantity or expression (b² – 4ac) under the square root sign. We say this because the root of a negative number can’t be any real number. Say x² = -1 is a quadratic equation. There is no real number whose square is negative. Therefore for this equation, there are no real number solutions.

Hence, the expression (b² – 4ac) is called the discriminant of the quadratic equation ax² + bx + c = 0. Its value determines the nature of roots as we shall see. Depending on the values of the discriminant, we shall see some cases about the nature of roots of different quadratic equations