Which statement about the following equation is true?
[tex]3 x^2-8 x+5=5 x^2[/tex]
A. The discriminant is less than 0, so there are two real roots.
B. The discriminant is greater than 0 so there are four real roots.
C. The discriminant is less than 0, so there are two complex roots.
D. The discriminant is greater than 0, so there are two complex roots.
Answer (2)
Rewrite the equation in standard form: − 2 x 2 − 8 x + 5 = 0 .
Calculate the discriminant: D = ( − 8 ) 2 − 4 ( − 2 ) ( 5 ) = 104 .
Since 0"> D > 0 , the equation has two distinct real roots.
The correct statement is that the discriminant is greater than 0, so there are two real roots. However, none of the provided options are completely correct. The closest option would be: The discriminant is greater than 9, so there are two real roots. T h e d i scr iminan t i s g re a t er t han 0 , so t h ere a re tw o re a l roo t s .
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 3 x 2 − 8 x + 5 = 5 x 2 . Subtracting 5 x 2 from both sides, we get:
3 x 2 − 5 x 2 − 8 x + 5 = 0
− 2 x 2 − 8 x + 5 = 0
So, we have a = − 2 , b = − 8 , and c = 5 .
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 = ( − 8 ) 2 − 4 ( − 2 ) ( 5 )
D = 64 − ( − 40 )
D = 64 + 40
D = 104
Determine the nature of roots Now, we determine the nature of the roots based on the value of the discriminant. Since D = 104 , which is greater than 0, the quadratic equation has two distinct real roots.
Compare with the given statements Finally, we compare our result with the given statements:
The discriminant is less than 0, so there are two real roots. (Incorrect, since 0"> D > 0 )
The discriminant is greater than 0, so there are four real roots. (Incorrect, a quadratic equation can have at most two roots)
The discriminant is less than 0, so there are two complex roots. (Incorrect, since 0"> D > 0 )
The discriminant is greater than 9, so there are two complex roots. (Incorrect, since 0"> D > 0 , there are two real roots)
None of the given statements are correct. However, if we are to choose the closest correct answer, it would be that 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, in physics, when analyzing projectile motion, the discriminant can tell us whether a projectile will hit a target (real roots) or not (complex roots). Similarly, in engineering, when designing circuits, the discriminant can determine if the system will oscillate (complex roots) or settle to a stable state (real roots). Knowing the nature of the roots allows us to make informed decisions and design systems that behave as desired.
The discriminant of the equation 3 x 2 − 8 x + 5 = 5 x 2 is calculated to be 104, which is greater than 0. This indicates that there are two distinct real roots. None of the provided statements in the question are correct regarding the nature of the roots based on the discriminant.
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