Identify the equivalent expression in the equation 1/x^2−x+1?x=5/x^2−x and demonstrate multiplying by the common denominator.

1 answer

To solve the equation \( \frac{1}{x^2 - x + 1} = \frac{x}{x^2 - x} \) and demonstrate multiplying by the common denominator, we first need to identify the common denominator.

The two denominators in the equation are:

  1. \( x^2 - x + 1 \)
  2. \( x^2 - x \)

The common denominator for both is the product of these two expressions. Thus, the common denominator \( D \) is:

\[ D = (x^2 - x + 1)(x^2 - x) \]

Next, we multiply both sides of the equation by this common denominator to eliminate the fractions:

\[ D \cdot \frac{1}{x^2 - x + 1} = D \cdot \frac{x}{x^2 - x} \]

Now, simplifying the left side:

\[ D \cdot \frac{1}{x^2 - x + 1} = (x^2 - x)(1) = x^2 - x \]

For the right side:

\[ D \cdot \frac{x}{x^2 - x} = (x^2 - x + 1)(x) \]

Now our equation simplifies to:

\[ x^2 - x = (x^2 - x + 1)(x) \]

Expanding the right side gives:

\[ x^2 - x = x^3 - x^2 + x \]

Now, we have a polynomial equation:

\[ x^2 - x = x^3 - x^2 + x \]

To bring everything to one side, we rearrange it:

\[ 0 = x^3 - 2x^2 + 2x \]

We can factor this polynomial to make it easier to solve:

\[ 0 = x(x^2 - 2x + 2) \]

The factored form indicates one solution is \( x = 0 \). To find the other solutions, we need to consider the quadratic:

\[ x^2 - 2x + 2 = 0 \]

Calculating the discriminant:

\[ D = b^2 - 4ac = (-2)^2 - 4(1)(2) = 4 - 8 = -4 \]

Since the discriminant is negative, there are no real solutions for this quadratic.

Therefore, the only solution to the original equation is:

\[ \boxed{0} \]