Question

determine the discriminant for 6x² - 3x + 3 = 0
based on the discriminant, what is the nature of the solution(s) for the equation?
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Explanation:

Step1: Identify the coefficients

For the quadratic equation \(ax^{2}+bx + c=0\), in the equation \(6x^{2}-3x + 3=0\), \(a = 6\), \(b=-3\), \(c = 3\).

Step2: Calculate the discriminant

The discriminant formula is \(\Delta=b^{2}-4ac\).
Substitute \(a = 6\), \(b=-3\), \(c = 3\) into the formula:

$$
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$$

Step3: Determine the nature of the solutions

If \(\Delta>0\), two real solutions; if \(\Delta = 0\), one real solution; if \(\Delta<0\), two complex solutions.
Since \(\Delta=-63<0\), the equation has two complex solutions.

Answer:

The discriminant is \(-63\) and the nature of the solutions is two complex solutions.