Q. \(x^{2}-4x+3=0\)

We want to solve the quadratic equation

\[x^2 – 4x + 3 = 0.\]

Step 1: Factor the quadratic.

For a quadratic of the form \(x^2 + bx + c\), we look for two numbers that:

1. Multiply to \(c\).
2. Add to \(b\).

Here, \(b = -4\) and \(c = 3\). We need two numbers that multiply to \(3\) and add to \(-4\).

The numbers \(-1\) and \(-3\) satisfy this because:

\[(-1)(-3) = 3,\]

\[(-1) + (-3) = -4.\]

So we can factor the polynomial:

\[x^2 – 4x + 3 = (x – 1)(x – 3).\]

Step 2: Use the zero product property.

If

\[(x – 1)(x – 3) = 0,\]

then at least one of the factors must equal \(0\).

Step 3: Set each factor equal to zero and solve.

First factor:

\[x – 1 = 0 \quad \Rightarrow \quad x = 1.\]

Second factor:

\[x – 3 = 0 \quad \Rightarrow \quad x = 3.\]

Final Answer:

The solutions to \(x^2 – 4x + 3 = 0\) are

\[x = 1 \quad \text{or} \quad x = 3.\]