What is the value of the discriminant of the quadratic equation $-2 x^2=-8 x+8$, and what does its value mean about the number of real number solutions the equation has?
A. The discriminant is equal to 0, which means the equation has no real number solutions.
B. The discriminant is equal to 0, which means the equation has one real number solution.
C. The discriminant is equal to 128, which means the equation has no real number solutions.
D. The discriminant is equal to 128, which means the equation has two real number solutions.
Answer (2)
Rewrite the quadratic equation in standard form: − 2 x 2 + 8 x − 8 = 0 .
Identify the coefficients: a = − 2 , b = 8 , c = − 8 .
Calculate the discriminant: D = b 2 − 4 a c = 8 2 − 4 ( − 2 ) ( − 8 ) = 0 .
Since the discriminant is 0, the equation has one real solution: 0 .
Explanation
Understanding the Problem We are given the quadratic equation − 2 x 2 = − 8 x + 8 . Our goal is to find the discriminant and determine how many real solutions the equation has.
Rewriting the Equation First, we need to rewrite the equation in the standard form a x 2 + b x + c = 0 . Adding 8 x and − 8 to both sides of the equation, we get: − 2 x 2 + 8 x − 8 = 0
Identifying Coefficients Now, we can identify the coefficients: a = − 2 , b = 8 , and c = − 8 .
Calculating the Discriminant The discriminant, denoted by D , is given by the formula D = b 2 − 4 a c . Substituting the values of a , b , and c , we have: D = ( 8 ) 2 − 4 ( − 2 ) ( − 8 ) = 64 − 64 = 0
Determining the Number of Real Solutions Since the discriminant D = 0 , the quadratic equation has exactly one real solution (a repeated root).
Examples
Understanding the discriminant helps us predict the nature of solutions in various real-world scenarios. For instance, in engineering, when designing a bridge, the equation describing the load distribution might be quadratic. If the discriminant is negative, it indicates the bridge design is stable under all load conditions. If it's zero, it means the bridge is at a critical stability point, and if it's positive, the design might be unstable and require modifications. This ensures safety and efficiency in structural designs.
The discriminant of the quadratic equation − 2 x 2 = − 8 x + 8 is calculated to be 0, indicating there is one real solution. Thus, the answer is option B. This means the graph of the quadratic touches the x-axis at exactly one point.
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