Question

Question content area topPart 1Evaluate the discriminant for the equation. Determine the number of real solutions.6 x squared plus x plus 8 equals 0Question content area bottomPart 1What is the discriminant of the quadratic​ equation?

Answer 1

To evaluate the discriminant for the quadratic equation \(6x^2 + x + 8 = 0\), we can use the formula for the discriminant:

\[
D = b^2 - 4ac
\]

In this equation, the coefficients are:
- \(a = 6\)
- \(b = 1\)
- \(c = 8\)

Now plug these values into the formula for the discriminant:

\[
D = (1)^2 - 4(6)(8)
\]

Calculating this step by step:

1. Calculate \(b^2\):
\[
1^2 = 1
\]

2. Calculate \(4ac\):
\[
4 \cdot 6 \cdot 8 = 192
\]

3. Now, substitute these values back into the discriminant equation:
\[
D = 1 - 192 = -191
\]

Therefore, the discriminant \(D\) is \(-191\).

The number of real solutions can be determined based on the value of the discriminant:
- If \(D > 0\), there are 2 distinct real solutions.
- If \(D = 0\), there is exactly 1 real solution (a repeated root).
- If \(D < 0\), there are no real solutions (the solutions are complex).

Since our discriminant is \(-191\), which is less than 0, we conclude that:

**There are no real solutions for the equation \(6x^2 + x + 8 = 0\).**