Question

find the discriminant.
\\(6t^{2} + 9t + 1 = 0\\)
what type of solutions does the equation have?
one real solution
two real solutions
two complex (non - real) solutions

Explanation:

Step1: Recall discriminant formula

For a quadratic equation \(at^2 + bt + c = 0\), the discriminant \(D\) is given by \(D = b^2 - 4ac\).

Step2: Identify coefficients

In the equation \(6t^2 + 9t + 1 = 0\), we have \(a = 6\), \(b = 9\), and \(c = 1\).

Step3: Calculate discriminant

Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula:
\[

$$\begin{align*} D&=9^2 - 4\times6\times1\\ &= 81 - 24\\ &= 57 \end{align*}$$

\]

Step4: Determine solution type

Since the discriminant \(D = 57>0\), the quadratic equation has two distinct real solutions.

Answer:

The discriminant is \(57\), and the equation has two real solutions.