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.
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.
