Which statement about the following equation is true?
[tex]2 x^2-9 x+2=-1[/tex]
A. The discriminant is less than 0, so there are two real roots.
B. The discriminant is less than 0, so there are two complex roots.
C. The discriminant is greater than 0, so there are two real roots.
D. The discriminant is greater than 0, so there are two complex roots.
Answer (2)
The given quadratic equation simplifies to 2 x 2 − 9 x + 3 = 0 . The discriminant is calculated as D = 57 , which is greater than 0, indicating there are two real roots. Therefore, the correct option is C.
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Rewrite the given equation in the standard quadratic form: 2 x 2 − 9 x + 3 = 0 .
Identify the coefficients: a = 2 , b = − 9 , and c = 3 .
Calculate the discriminant: D = b 2 − 4 a c = ( − 9 ) 2 − 4 ( 2 ) ( 3 ) = 57 .
Since the discriminant 0"> D > 0 , the equation has two real roots. The correct statement is: The discriminant is greater than 0, so there are two real roots. $\boxed{The discriminant is greater than 0 , so there are Iwo real roots.}
Explanation
Understanding the Problem We are given the quadratic equation 2 x 2 − 9 x + 2 = − 1 . Our goal is to determine the nature of its roots by analyzing the discriminant.
Rewriting the Equation First, we need to rewrite the equation in the standard quadratic form a x 2 + b x + c = 0 . Adding 1 to both sides of the equation, we get: 2 x 2 − 9 x + 2 + 1 = − 1 + 1 2 x 2 − 9 x + 3 = 0
Identifying Coefficients Now, we identify the coefficients a , b , and c from the standard form a x 2 + b x + c = 0 . In our equation, we have: a = 2 b = − 9 c = 3
Calculating the Discriminant Next, we calculate the discriminant, D , using the formula D = b 2 − 4 a c . Substituting the values of a , b , and c , we get: D = ( − 9 ) 2 − 4 ( 2 ) ( 3 ) D = 81 − 24 D = 57
Analyzing the Discriminant Now, we analyze the discriminant. Since D = 57 , which is greater than 0, the quadratic equation has two distinct real roots.
Conclusion Therefore, the correct statement is: The discriminant is greater than 0, so there are two real roots.
Examples
Understanding the discriminant helps in various real-world applications. For example, in engineering, when designing a bridge, the equation describing the load and stress must have real roots to ensure the bridge's stability. If the discriminant is negative, it indicates that the equation has complex roots, which means the design is not feasible and the bridge would be unstable. Similarly, in physics, analyzing the discriminant of equations describing projectile motion helps determine if a projectile will hit a target or not. These examples highlight the importance of understanding the nature of roots in practical scenarios.
