Question

Simplify the expression. $$\frac{x^{2}-4 x+3}{2 x} \div \frac{x-1}{2}$$

Short answer

The simplified form of the original expression is \( \frac{x - 3}{x} \)

Step-by-step solution

Rewrite the Problem

Rewrite the complex fraction expression as a multiplication problem. Rewrite the division as multiplication by taking the reciprocal of the divisor. The expression should then read as follows: \( \frac{x^{2}-4 x+3}{2 x} \times \frac{2}{x-1} \)

Simplify the Complex Fraction

Now simplify the smaller fractions before multiplying them. Here, the 2s cancel out, this leaves us with \( \frac{x^{2}-4 x+3}{x} \times \frac{1}{x-1} \)

Factor the Polynomial

Factor the polynomial in the numerator. \( x^{2} - 4x + 3 \) can be factored to \( (x-3)(x-1) \). So the expression becomes \( \frac{(x - 3)(x - 1)}{x} \times \frac{1}{x - 1} \)

Simplify the Expression

Cancel out the common factors in the numerator and denominator. Here \( x - 1 \) is a common factor, so the expression simplifies to \( \frac{x - 3}{x} \)

Check your Result

Ensure that no further simplification is possible in \( \frac{x - 3}{x} \). This is the simplified form of the original expression