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 discriminant of the equation 2 x 2 − 9 x + 3 = 0 is calculated to be 57, which is greater than 0. Therefore, this means the equation has two distinct real roots. The correct option is C: The discriminant is greater than 0, so there are two real roots.
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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 , the equation has 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 . To rewrite it in the standard form, we add 1 to both sides: 2 x 2 − 9 x + 2 + 1 = − 1 + 1 2 x 2 − 9 x + 3 = 0
Identify coefficients Now, we identify the coefficients a , b , and c in the quadratic equation 2 x 2 − 9 x + 3 = 0 . We have: a = 2 b = − 9 c = 3
Calculate the discriminant Next, we calculate the discriminant, denoted by 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
Determine the sign of the discriminant Now, we determine the sign of the discriminant. Since D = 57 , we have 0"> D > 0 .
Conclusion Since the discriminant D is greater than 0, the quadratic equation has two distinct real roots. Therefore, the correct statement is: The discriminant is greater than 0, so there are two real roots.
Examples
Understanding the discriminant helps us determine the nature of the roots of a quadratic equation without actually solving for the roots. For example, in engineering, when designing a bridge, the equation describing the load distribution might be quadratic. Knowing whether the equation has real roots tells engineers if there are specific load points where stress is zero, which is crucial for safety. Similarly, in physics, analyzing projectile motion involves quadratic equations, and the discriminant helps determine if a projectile will hit a target (real roots) or not (complex roots).
