Certainly! Let's solve the quadratic equation [tex]\( x^2 - 4x + 3 = 0 \)[/tex] step-by-step.
A quadratic equation is generally of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. Here, we can see that:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = 3 \)[/tex]
### Step 1: Calculate the Discriminant
The discriminant [tex]\( \Delta \)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the formula:
[tex]\[ \Delta = (-4)^2 - 4 \cdot 1 \cdot 3 \][/tex]
[tex]\[ \Delta = 16 - 12 \][/tex]
[tex]\[ \Delta = 4 \][/tex]
### Step 2: Analyze the Discriminant
Since the discriminant [tex]\( \Delta = 4 \)[/tex] is positive, we know that the equation has two distinct real roots.
### Step 3: Find the Roots
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex] into the formula:
[tex]\[ x = \frac{-(-4) \pm \sqrt{4}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{4 \pm 2}{2} \][/tex]
This gives us two solutions:
1. When using the positive square root:
[tex]\[ x_1 = \frac{4 + 2}{2} = \frac{6}{2} = 3 \][/tex]
2. When using the negative square root:
[tex]\[ x_2 = \frac{4 - 2}{2} = \frac{2}{2} = 1 \][/tex]
### Conclusion
The discriminant is [tex]\( 4 \)[/tex].
The roots of the equation [tex]\( x^2 - 4x + 3 = 0 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = 1 \)[/tex].
So, the solution is:
[tex]\[ \Delta = 4,\ x = 3, \ x = 1 \][/tex]
