What is the relationship between the coefficients of the function f(x) = 5x² + 36x + 54 and its discriminant?

The discriminant of a quadratic function, given by the formula (b^2 - 4ac) where (a), (b), and (c) are the coefficients of the function (f(x) = ax^2 + bx + c), plays a critical role in determining the nature of the roots of the quadratic equation. In the case of the function (f(x) = 5x^2 + 36x + 54), the coefficients are (a = 5), (b = 36), and (c = 54).

By calculating the discriminant:

[

D = b^2 - 4ac = 36^2 - 4(5)(54) = 1296 - 1080 = 216

]

The value of the discriminant helps us understand the number and type of the quadratic's roots. Specifically:

• If the discriminant is positive (as it is in this case), the quadratic has two distinct real roots.

• If the discriminant is zero, there is exactly one real root, or the roots are repeated.

• If the discriminant is negative, the roots are complex (not real).

Thus, the