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 two 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.

The quadratic equation 3 x 2 − 8 x + 5 = 5 x 2 can be rewritten as 2 x 2 + 8 x − 5 = 0 . Calculating the discriminant gives D = 104 , which is greater than 0, indicating there are two distinct real roots. Therefore, the correct choice is: the discriminant is greater than 0, so there are two real roots.
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Answered by Anonymous | 2025-07-04

Rewrite the given equation in the standard quadratic form: 2 x 2 + 8 x − 5 = 0 .
Identify the coefficients: a = 2 , b = 8 , and c = − 5 .
Calculate the discriminant: D = b 2 − 4 a c = 8 2 − 4 ( 2 ) ( − 5 ) = 104 .
Since the discriminant is greater than 0, the equation has two real roots: The discriminant is greater than 0, so there are two real roots. ​

Explanation

Understanding the Problem We are given the equation 3 x 2 − 8 x + 5 = 5 x 2 . 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, which is a x 2 + b x + c = 0 . Subtracting 3 x 2 − 8 x + 5 from both sides of the equation 5 x 2 = 3 x 2 − 8 x + 5 gives us:


5 x 2 − ( 3 x 2 − 8 x + 5 ) = 0 5 x 2 − 3 x 2 + 8 x − 5 = 0 2 x 2 + 8 x − 5 = 0

Identifying Coefficients Now, we can identify the coefficients: a = 2 , b = 8 , and c = − 5 .

Calculating the Discriminant Next, we calculate the discriminant, denoted by D , using the formula D = b 2 − 4 a c . Plugging in the values of a , b , and c , we get:


D = ( 8 ) 2 − 4 ( 2 ) ( − 5 ) D = 64 − ( − 40 ) D = 64 + 40 D = 104

Determining the Nature of Roots Since the discriminant D = 104 is greater than 0, the quadratic equation has two distinct real roots.

Final Answer Therefore, the correct statement 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 instance, if you're designing a bridge and modeling its structure with a quadratic equation, the discriminant can tell you whether the supports will experience real stress points or if the model predicts a more stable, complex distribution of forces. Similarly, in projectile motion, the discriminant can determine if a ball will actually hit the ground (real roots) or if the trajectory is purely theoretical (complex roots). These applications highlight how knowing about the discriminant can provide valuable insights in engineering and physics.

Answered by GinnyAnswer | 2025-07-04