Which statement about the following equation is true?
[tex]$2 x^2-9 x+2=-1$[/tex]
A. The discriminant is greater than 0, so there are two complex 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 less than 0, so there are two real roots.
Answer (1)
Rewrite the equation in standard form: 2 x 2 − 9 x + 3 = 0 .
Identify the coefficients: a = 2 , b = − 9 , c = 3 .
Calculate the discriminant: D = ( − 9 ) 2 − 4 ( 2 ) ( 3 ) = 57 .
Since 0"> D > 0 , there are two real roots: The discriminant is greater than 0, so there are two real roots.
Explanation
Rewrite the equation First, we need to rewrite the given equation in the standard quadratic form, which is a x 2 + b x + c = 0 . The given equation is 2 x 2 − 9 x + 2 = − 1 . Adding 1 to both sides, we get 2 x 2 − 9 x + 3 = 0 .
Identify coefficients Now, we identify the coefficients a , b , and c . In this case, a = 2 , b = − 9 , and c = 3 .
Calculate the discriminant Next, we calculate the discriminant, which is given by the formula D = b 2 − 4 a c . Substituting the values of a , b , and c , we have: D = ( − 9 ) 2 − 4 ( 2 ) ( 3 ) = 81 − 24 = 57
Determine the nature of roots Now, we determine the nature of the roots based on the discriminant. Since 0"> D = 57 > 0 , the quadratic equation has two distinct real roots.
Final Answer Therefore, the statement that is true about the equation is: The discriminant is greater than 0, so there are two real roots.
Examples
Understanding the discriminant helps us predict the type of solutions we'll get when solving quadratic equations. For example, if you're designing a bridge and modeling its structure with a quadratic equation, knowing whether the roots are real tells you if the bridge design is physically possible. If the roots are complex, it indicates the design parameters need adjustment to achieve a stable, real-world structure. Similarly, in projectile motion, the discriminant can tell you if a ball will actually hit the ground (real roots) or if the model predicts it will never land (complex roots, indicating the model might be oversimplified).
