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Which statement about the following equation is true?

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

A. The discriminant is greater than 0, so there are two real roots.
B. The discriminant is less than 0, so there are two real roots.
C. The discriminant is greater than 0, so there are two complex roots.
D. The discriminant is less than 0, so there are two complex roots.

Sagot :

GinnyAnsweridn
To determine the correct statement about the given quadratic equation [tex]\( 3x^2 - 8x + 5 = 5x^2 \)[/tex], we will follow a step-by-step approach:

1. Rewrite the Equation:
First, we need to bring the equation to the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex].
Given: [tex]\( 3x^2 - 8x + 5 = 5x^2 \)[/tex]
Subtract [tex]\( 5x^2 \)[/tex] from both sides to combine like terms:
[tex]\[ 3x^2 - 8x + 5 - 5x^2 = 0 \][/tex]
This simplifies to:
[tex]\[ -2x^2 - 8x + 5 = 0 \][/tex]

2. Identify the Coefficients:
Now, we need to identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
[tex]\[ a = -2, \quad b = -8, \quad c = 5 \][/tex]

3. Calculate the Discriminant:
The discriminant ([tex]\(D\)[/tex]) of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ D = b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula, we get:
[tex]\[ D = (-8)^2 - 4(-2)(5) \][/tex]
[tex]\[ D = 64 + 40 \][/tex]
[tex]\[ D = 104 \][/tex]

4. Interpret the Discriminant:
A discriminant greater than 0 indicates that the quadratic equation has two distinct real roots.
[tex]\[ D = 104 > 0 \][/tex]

Based on the result of our calculation, the discriminant is greater than 0, so the true statement about the given equation [tex]\( 3x^2 - 8x + 5 = 5x^2 \)[/tex] is:

\- "The discriminant is greater than 0, so there are two real roots."
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