Question

find the discriminant.
\\(5p^2 + 9p + 3 = 0\\)
what type of solutions does the equation have?
one real solution
two real solutions
two complex (non - real) solutions

Explanation:

Step1: Recall the discriminant formula

For a quadratic equation \(ax^2 + bx + c = 0\), the discriminant \(D\) is given by \(D = b^2 - 4ac\).
In the equation \(5p^2 + 9p + 3 = 0\), we have \(a = 5\), \(b = 9\), and \(c = 3\).

Step2: Calculate the discriminant

Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula:
\(D = 9^2 - 4\times5\times3\)
\(= 81 - 60\)
\(= 21\)

Step3: Determine the type of solutions

Since the discriminant \(D = 21>0\), a quadratic equation with a positive discriminant has two distinct real solutions.

Answer:

The discriminant is \(21\) and the equation has two real solutions.