What is the value of the discriminant, [tex]$b^2-4 a c$[/tex], for the quadratic equation [tex]$0=x^2-4 x+5$[/tex], and what does it mean about the number of real solutions the equation has?
A. The discriminant is -4, so the equation has 2 real solutions.
B. The discriminant is -4, so the equation has no real solutions.
C. The discriminant is 35, so the equation has 2 real solutions.
D. The discriminant is 35, so the equation has no real solutions.
Answer (2)
Identify the coefficients: a = 1 , b = − 4 , c = 5 .
Calculate the discriminant: b 2 − 4 a c = ( − 4 ) 2 − 4 ( 1 ) ( 5 ) = 16 − 20 = − 4 .
Since the discriminant is negative ( − 4 < 0 ), the quadratic equation has no real solutions.
The discriminant is − 4 , so the equation has no real solutions: The discriminant is -4, so the equation has no real solutions.
Explanation
Understanding the Problem We are given the quadratic equation 0 = x 2 − 4 x + 5 . Our goal is to find the discriminant, b 2 − 4 a c , and determine how many real solutions the equation has based on the discriminant's value.
Identifying Coefficients First, we need to identify the coefficients a , b , and c in the quadratic equation. Comparing 0 = x 2 − 4 x + 5 to the standard form a x 2 + b x + c = 0 , we have a = 1 , b = − 4 , and c = 5 .
Calculating the Discriminant Now, we calculate the discriminant using the formula b 2 − 4 a c . Substituting the values we found, we get: ( − 4 ) 2 − 4 ( 1 ) ( 5 ) = 16 − 20 = − 4.
Determining the Number of Real Solutions The discriminant is − 4 . Since the discriminant is negative, the quadratic equation has no real solutions. This is because the square root of a negative number is not a real number, and the quadratic formula involves taking the square root of the discriminant.
Final Answer Therefore, the value of the discriminant is − 4 , and the equation has no real solutions.
Examples
Understanding the discriminant helps us predict the nature of solutions in various real-world scenarios. For example, when designing a bridge, engineers use quadratic equations to model the structure's stability. If the discriminant is negative, it indicates that the design might not be stable under certain conditions, prompting a redesign. Similarly, in projectile motion, a negative discriminant could mean that a projectile will never reach a certain height, influencing launch parameters.
The discriminant of the quadratic equation 0 = x 2 − 4 x + 5 is calculated to be − 4 . This negative value indicates that the equation has no real solutions. Therefore, the correct answer is B.
;
