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In any quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant plays a crucial role in determining the nature of the roots. The discriminant (\( \text{D} \)) is found using the formula \[ D = b^2 - 4ac \]. It is the expression under the square root in the quadratic formula.

Here are the key points to understand:

• If D > 0, the equation has two distinct real roots.
• If D = 0, the equation has exactly one real root (a repeated root).
• If D < 0, the equation has no real roots but two complex (imaginary) roots.

In our example, the coefficients are a = 2, b = 3, and c = 3. Substituting these into the discriminant formula gives: \( D = 3^2 - 4(2)(3) = 9 - 24 = -15 \).

Since \( D < 0 \), the quadratic equation \( 2x^2 + 3x + 3 = 0 \) has two complex roots.

Imaginary numbers come into play when the discriminant of a quadratic equation is negative. The square root of a negative number is not a real number. Instead, it is an imaginary number. The imaginary unit is denoted by the symbol \( i \), and it is defined as the square root of -1, i.e., \( i = \sqrt{{-1}}\).

When we deal with the square root of a negative discriminant, we write it in terms of \( i \). For our example, \( D = -15 \). Therefore, \( \sqrt{{-15}} = \sqrt{{-1 \cdot 15}} = i \sqrt{{15}} \).

This means that the quadratic equation has complex roots, which we express as \(-3 \pm i \sqrt{{15}} \)/4. Complex roots always come in conjugate pairs, such as \( a + bi \) and \( a - bi \), ensuring that the imaginary parts cancel out when added together.

The quadratic formula is used to solve any quadratic equation of the form \( ax^2 + bx + c = 0 \). To remember, the formula is: \[ x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{{2a}} \].

Here are the steps to solve using the quadratic formula:

• Identify the coefficients a, b, and c in the equation.
• Substitute the coefficients into the quadratic formula.
• Calculate the discriminant (\( D = b^2 - 4ac \)).
• Determine the nature of the roots using the discriminant.
• Simplify the expression to find the roots.

In our problem \( 2x^2 + 3x + 3 = 0 \):
- Coefficients are a = 2, b = 3, c = 3.
- Substituting these into the quadratic formula we get: \[ x = \frac{{-3 \pm \sqrt{3^2 - 4(2)(3)}}}{{2(2)}} = \frac{{-3 \pm \sqrt{-15}}}{{4}} \].
- Simplifying, we find:
\[ x = \frac{{-3 \pm i \sqrt{15}}}{{4}} \]
The two solutions are: \(\frac{{-3 + i \sqrt{15}}}{{4}} \) and \(\frac{{-3 - i \sqrt{15}}}{{4}} \).
These are the complex roots of the equation.