91Ó°ÊÓ

The discriminant is an important part of the quadratic formula. It gives us crucial information about the nature of the roots of a quadratic equation. The discriminant is represented by the expression \( D = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the coefficients from the standard form of a quadratic equation \( ax^2 + bx + c = 0 \).

Here's why the discriminant is so useful:

• If \( D > 0 \), the quadratic equation has two distinct real roots.
• If \( D = 0 \), the equation has exactly one real root, also known as a repeated or double root.
• If \( D < 0 \), there are no real roots, but instead, two complex roots exist.

In our exercise, the discriminant was calculated as \( -18 \), indicating that our equation has complex roots. Calculating the discriminant gives us an early clue about the solution's nature.

When a quadratic equation has a negative discriminant, it means the roots are not real. Instead, they are complex numbers. In a complex number, we have a real part and an imaginary part, denoted by \( i \), which is the square root of \( -1 \).

For the quadratic equation from our exercise, the complex roots can be found using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Since \( \sqrt{-18} \) involves an imaginary unit, we rewrite it as \( 3i\sqrt{2} \). Thus, the solutions are expressed as \( 2 \pm i\sqrt{2} \).

These complex solutions indicate that the parabola described by the quadratic equation does not intersect the x-axis. Instead, the solutions exist in a complex plane where each entails a combination of real and imaginary parts.

Coefficients in a quadratic equation are crucial because they define the shape and position of the parabola represented by the equation. The standard form of any quadratic equation is \( ax^2 + bx + c = 0 \). Each of the values \( a \), \( b \), and \( c \) determines a different aspect of the parabola:

• \( a \): Affects the width and the direction of the parabola. If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.
• \( b \): Influences the placement along the x-axis by affecting the symmetry axis of the parabola.
• \( c \): This is the y-intercept, the point where the parabola crosses the y-axis.

In our exercise, the coefficients were \( a = \frac{3}{2} \), \( b = -6 \), and \( c = 9 \). Identifying and using these coefficients was key to applying both the discriminant and the quadratic formula successfully to solve the equation.