Question

find the discriminant.
\\(6t^2 + 8t + 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 \(at^2 + bt + c = 0\), the discriminant \(D\) is given by \(D = b^2 - 4ac\).
In the equation \(6t^2 + 8t + 3 = 0\), we have \(a = 6\), \(b = 8\), and \(c = 3\).

Step2: Calculate the discriminant

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

$$\begin{align*} D&=8^2 - 4\times6\times3\\ &=64 - 72\\ &=-8 \end{align*}$$

\]

Step3: Determine the type of solutions

• If \(D>0\), the equation has two distinct real solutions.
• If \(D = 0\), the equation has one real solution (a repeated root).
• If \(D<0\), the equation has two complex (non - real) solutions.

Since \(D=-8<0\), the equation \(6t^2 + 8t + 3 = 0\) has two complex (non - real) solutions.

Answer:

The discriminant is \(-8\). The equation has two complex (non - real) solutions.