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 given quadratic equation can be rearranged to give a standard form where the discriminant is calculated to be 104. Since the discriminant is greater than 0, the equation has two real roots. Thus, option B is correct.
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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 , c = − 5 .
Calculate the discriminant: D = b 2 − 4 a c = 8 2 − 4 ( 2 ) ( − 5 ) = 104 .
Since 0"> D > 0 , the equation has two real roots. The answer is: 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 a x 2 + b x + c = 0 . Subtracting 3 x 2 − 8 x + 5 from both sides, we get:


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 D using the formula D = b 2 − 4 a c :


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.

Conclusion 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, in physics, when modeling projectile motion, the discriminant can tell us whether a projectile will hit a target (two real roots), just graze it (one real root), or miss it entirely (two complex roots). Similarly, in engineering, when designing a bridge, analyzing the discriminant of equations related to structural stability can determine if the bridge will withstand certain loads or if modifications are needed to ensure its integrity. This concept is also used in economics to model supply and demand curves and determine equilibrium points.

Answered by GinnyAnswer | 2025-07-04