To find the other root of the quadratic equation [tex]\(x^2 - 4x + 3 = 0\)[/tex] given that one root is 1, we can use the properties of quadratic equations and the relationships between their roots and coefficients.
A quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex] has roots that satisfy the following properties:
1. The sum of the roots ([tex]\(r_1 + r_2\)[/tex]) is given by [tex]\(-\frac{b}{a}\)[/tex].
2. The product of the roots ([tex]\(r_1 \cdot r_2\)[/tex]) is given by [tex]\(\frac{c}{a}\)[/tex].
Given equation: [tex]\(x^2 - 4x + 3 = 0\)[/tex]
Here:
[tex]\(a = 1\)[/tex]
[tex]\(b = -4\)[/tex]
[tex]\(c = 3\)[/tex]
We are given that one root, [tex]\(r_1\)[/tex], is [tex]\(1\)[/tex]. We need to find the other root, [tex]\(r_2\)[/tex].
1. Sum of the Roots:
The sum of the roots is:
[tex]\[ r_1 + r_2 = -\frac{b}{a} = -\frac{-4}{1} = 4 \][/tex]
Since [tex]\(r_1 = 1\)[/tex]:
[tex]\[ 1 + r_2 = 4 \][/tex]
Solving for [tex]\(r_2\)[/tex]:
[tex]\[ r_2 = 4 - 1 \][/tex]
[tex]\[ r_2 = 3 \][/tex]
2. Verification with Product of the Roots:
The product of the roots is:
[tex]\[ r_1 \cdot r_2 = \frac{c}{a} = \frac{3}{1} = 3 \][/tex]
Substituting [tex]\(r_1 = 1\)[/tex] and [tex]\(r_2 = 3\)[/tex]:
[tex]\[ 1 \cdot 3 = 3 \][/tex]
Since the product of [tex]\(1\)[/tex] and [tex]\(3\)[/tex] equals [tex]\(3\)[/tex], our calculated root [tex]\(r_2 = 3\)[/tex] satisfies both the sum and product properties.
Thus, the other root of the quadratic equation [tex]\(x^2 - 4x + 3 = 0\)[/tex] is:
[tex]\[ \boxed{3} \][/tex]
So, the correct answer is:
(C) 3
