Certainly! Let's solve the given quadratic equation [tex]\[ x^2 - 5x + 4 = 0 \][/tex] and determine the nature of its roots.
### Step-by-Step Solution:
1. Identify the coefficients of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -5 \)[/tex]
- [tex]\( c = 4 \)[/tex]
2. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plugging in the coefficients:
[tex]\[
\Delta = (-5)^2 - 4 \cdot 1 \cdot 4 = 25 - 16 = 9
\][/tex]
3. Determine the nature of the roots based on the value of the discriminant [tex]\(\Delta\)[/tex]:
- If [tex]\(\Delta > 0\)[/tex], there are two distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], there is one real repeated root.
- If [tex]\(\Delta < 0\)[/tex], there are two complex roots.
Since [tex]\(\Delta = 9 \)[/tex], which is greater than zero, the quadratic equation [tex]\( x^2 - 5x + 4 = 0 \)[/tex] has:
[tex]\[
\boxed{\text{Two distinct real roots}}
\][/tex]
### Conclusion:
- The discriminant of the quadratic equation [tex]\( x^2 - 5x + 4 = 0 \)[/tex] is [tex]\( \Delta = 9 \)[/tex].
- The nature of the roots is: Two distinct real roots.
