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."
