(1) PLAY \( 8 x^{2}-4 x+1=0 \) (a) Find the discriminant.

To find the discriminant for the quadratic equation \(8x^{2}-4x+1=0\), we use the formula \(D = b^2 - 4ac\). Here, \(a = 8\), \(b = -4\), and \(c = 1\). Plugging in these values, we get: \[ D = (-4)^2 - 4(8)(1) = 16 - 32 = -16. \] So the discriminant is \(-16\). When the discriminant is negative, it tells us that there are no real solutions to the quadratic equation, which means the graph doesn't touch the x-axis. Instead, it intersects the y-axis at the value of \(c\), making it essential for understanding the nature of the roots! For a practical application of understanding discriminants, think of it as your guidance system. A positive discriminant means two distinct real roots—yay for intersections! A zero indicates one real root, or as some say, a double root—like a kiss goodbye where they just can’t let go! Meanwhile, a negative discriminant tells us there's a complex solution, something found often in physics and engineering problems dealing with oscillations and waves. It's like when life gives you imaginary numbers—it can be quite a ride!