Determine the number and type of solutions of the quadratic equation. \( -4 x^{2}-4 x-1=0 \) two imaginary solutions two real solutions one real solution one imaginary solution

To determine the number and type of solutions for the quadratic equation \( -4x^2 - 4x - 1 = 0 \), we can use the discriminant \( D = b^2 - 4ac \). Here, \( a = -4 \), \( b = -4 \), and \( c = -1 \). Calculating the discriminant gives: \[ D = (-4)^2 - 4(-4)(-1) = 16 - 16 = 0. \] Since the discriminant is equal to zero, the equation has exactly one real solution (a double root). In more relatable terms, think of trying to find the roots of this equation as searching for a hidden treasure. The discriminant helps you gauge how many treasures you might find: two treasures, one treasure, or a quest that leads you nowhere! Here, with a discriminant of zero, you’ve found that elusive treasure—just one, but it’s a special one since it’s a repeating root! Keep your treasure map handy; you’ll find that one golden spot where the equation touches the x-axis!