Question

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

Answer 1

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:

\[
\boxed{0}
\]