Solve the rational equation for x:  3x+x−3x=1x2 .(1 point)Responsesx=1x equals 1x=0x equals 0x=±1x equals plus or minus 1x=3

1 answer

To solve the rational equation \( \frac{3x + x - 3x}{1} = x^2 \), we first simplify the left-hand side:

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

So the equation becomes:

\[ x = x^2 \]

Now, we can rearrange it to set it to zero:

\[ x^2 - x = 0 \]

We can factor this equation:

\[ x(x - 1) = 0 \]

Setting each factor to zero gives:

  1. \( x = 0 \)
  2. \( x - 1 = 0 \) which leads to \( x = 1 \)

Thus, the solutions are:

\[ x = 0 \quad \text{and} \quad x = 1 \]

The possible responses include \( x = 1 \), \( x = 0 \), \( x = \pm 1 \), and \( x = 3 \). The correct solutions are:

\[ \boxed{x = 0} \quad \text{and} \quad \boxed{x = 1} \]