Is the valre of the discimimant, \( b^{2}-4 a c \), for the quadratic equation \( 0=x^{2}-4 x+5 \), and what does umber of real solutions the equation has? Tho disctiminant is -4 , so the equation has 2 real solutions, The disctiminant is \( =4 \), so the eqpation has no real solutions, The disctiminant is 35 , so the equation has 2 real solutions. The discrininant is 35 so the equalion las no real solutions.

To find the value of the discriminant for the quadratic equation \( 0 = x^{2} - 4x + 5 \), we start by calculating \( b^{2} - 4ac \). Here, \( a = 1 \), \( b = -4 \), and \( c = 5 \). Plugging in the values gives us \( (-4)^2 - 4(1)(5) = 16 - 20 = -4 \). A negative discriminant means that the equation has no real solutions; instead, you'll find two complex solutions. Did you know that the concept of the discriminant not only helps in solving quadratic equations but can also extend to polynomial equations of higher degrees? In the world of algebra, if the discriminant is zero, it indicates that the equation has one repeated real root, while any positive value means two distinct real solutions await. Dive into how imaginary numbers play a role in solving complex solutions—it’s like unlocking a secret door to a whole new realm of mathematics!