To find the roots of the quadratic equation \( x^2 - 4x + 3 = 0 \), we can use the quadratic formula, which is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, \( a = 1 \), \( b = -4 \), and \( c = 3 \).
1. First, identify the coefficients:
- \( a = 1 \)
- \( b = -4 \)
- \( c = 3 \)
2. Calculate the discriminant \( \Delta \) using the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values:
[tex]\[ \Delta = (-4)^2 - 4 \cdot 1 \cdot 3 = 16 - 12 = 4 \][/tex]
3. Since the discriminant is positive (\(\Delta = 4\)), the quadratic equation has two distinct real roots.
4. Now, use the quadratic formula to find the roots. The solutions are:
[tex]\[ x_1 = \frac{-b + \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x_2 = \frac{-b - \sqrt{\Delta}}{2a} \][/tex]
Substituting the values:
[tex]\[ x_1 = \frac{4 + \sqrt{4}}{2 \cdot 1} = \frac{4 + 2}{2} = \frac{6}{2} = 3 \][/tex]
[tex]\[ x_2 = \frac{4 - \sqrt{4}}{2 \cdot 1} = \frac{4 - 2}{2} = \frac{2}{2} = 1 \][/tex]
Therefore, the roots of the equation \( x^2 - 4x + 3 = 0 \) are \( x = 3 \) and \( x = 1 \).
In terms of coordinates, these roots can be represented as the points where the equation \( y = x^2 - 4x + 3 \) intersects the x-axis:
- The coordinates of the first root are \( (3, 0) \).
- The coordinates of the second root are \( (1, 0) \).
So, the coordinates of the roots are [tex]\( (3, 0) \)[/tex] and [tex]\( (1, 0) \)[/tex].
