what is the discriminant of 3x^2 + 2x +1 = 0?

In the Quadratic Formula, [tex]\displaystyle\mathsf{x\:=\:\frac{-b\pm\sqrt{b^2-4ac}}{2a}}[/tex], the expression under the radical symbol, "b² - 4ac" is the discriminant, which tells us the number of solutions of a given quadratic equation. When we substitute the values for a, b, and c of a given quadratic equation into the quadratic formula, we are essentially solving for its number of solutions.

Discriminant: b² - 4ac

If b² - 4ac > 0, then it means that the quadratic equation has two real solutions.

If b² - 4ac = 0, then it means that the quadratic equation has one real solution.

If b² - 4ac < 0, then it means that the quadratic equation has no real solutions.

Solution:

Given the quadratic equation, 3x² + 2x + 1 = 0, where a = 3, b = 2, and c = 1, let us substitute the values for a, b, and c into the discriminant to determine its number of solutions:

Discriminant: b² - 4ac = (2)² - 4(3)( 1 ) = 4 - 12 = - 8

Hence, the discriminant of 3x² + 2x + 1 = 0 is -8, which means that the given quadratic equation has no real solutions. This implies that the 3x² + 2x + 1 = 0 has no x-intercepts (because the graph does not cross the x-axis).

Attached is a screenshot of the graphed quadratic equation, 3x² + 2x + 1 = 0, where it shows that it does not cross the x-axis, proving that there are no real solutions.