Which statement about the following equation is true?

[tex]\[ 3x^2 - 8x + 5 = 5x^2 \][/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.



Answer :

To analyze the nature of the roots for the equation \( 3x^2 - 8x + 5 = 5x^2 \), we first need to rewrite it in the standard form of a quadratic equation: \( ax^2 + bx + c = 0 \).

Given:
[tex]\[ 3x^2 - 8x + 5 = 5x^2 \][/tex]

Let's rearrange the equation to bring all terms to one side:
[tex]\[ 3x^2 - 8x + 5 - 5x^2 = 0 \][/tex]

Combine like terms:
[tex]\[ -2x^2 - 8x + 5 = 0 \][/tex]

Multiply the entire equation by -1 to make \( ax^2 \) positive:
[tex]\[ 2x^2 - 8x + 5 = 0 \][/tex]

Here, we have:
[tex]\[ a = 2 \][/tex]
[tex]\[ b = -8 \][/tex]
[tex]\[ c = 5 \][/tex]

To determine the nature of the roots, we calculate the discriminant (\(\Delta\)) of the quadratic equation, which is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]

Substitute \( a \), \( b \), and \( c \) into the discriminant formula:
[tex]\[ \Delta = (-8)^2 - 4(2)(5) \][/tex]
[tex]\[ \Delta = 64 - 40 \][/tex]
[tex]\[ \Delta = 24 \][/tex]

Now, we interpret the discriminant:
- If \(\Delta > 0\), the equation has two distinct real roots.
- If \(\Delta = 0\), the equation has exactly one real root (repeated).
- If \(\Delta < 0\), the equation has two complex roots.

In this case, the discriminant (\(\Delta\)) is 24, which is greater than 0. Therefore, the quadratic equation \( 2x^2 - 8x + 5 = 0 \) has two distinct real roots.

Thus, the correct statement is:
The discriminant is greater than 0, so there are two real roots.