What is the discriminant of the quadratic equation \( 2x^{2} - 4x + 1 = 0 \)?

To find the discriminant of the quadratic equation \( 2x^{2} - 4x + 1 = 0 \), we can use the formula for the discriminant \( D = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the coefficients from the standard quadratic form \( ax^2 + bx + c = 0 \). In this equation, we have: - \( a = 2 \) - \( b = -4 \) - \( c = 1 \) Now, substituting these values into the discriminant formula, we get: \[ D = (-4)^2 - 4(2)(1) = 16 - 8 = 8 \] So, the discriminant of the quadratic equation \( 2x^{2} - 4x + 1 = 0 \) is \( 8 \). Now, let's consider the implications of that discriminant! A discriminant of \( 8 \) suggests that this quadratic equation has two distinct real roots. It's a sure sign that the graph of the parabola opens either upward or downward and crosses the x-axis at two different points. What a fun math party! If you're curious how this relates to real-world situations, consider projectile motion! When you throw an object into the air, the path it takes follows a quadratic equation. The discriminant can help you determine how many times the object reaches a certain height, such as when dropping a basketball from a height to see how many times it hits a specific point on the floor on the way down!