https://file.aiquickdraw.com/mathbot/file/17257884417320jbhppda.jpg

The image shows the equation to be solved:

\[
\frac{3}{x-3} = 1 - 2x
\]

Let's solve it step by step.

### Step 1: Eliminate the fraction
Multiply both sides of the equation by \( x - 3 \) to get rid of the fraction:

\[
3 = (1 - 2x)(x - 3)
\]

### Step 2: Expand the right-hand side
Distribute on the right-hand side:

\[
3 = (1)(x - 3) + (-2x)(x - 3)
\]
\[
3 = x - 3 - 2x^2 + 6x
\]

### Step 3: Combine like terms
Now, combine the like terms on the right-hand side:

\[
3 = -2x^2 + 7x - 3
\]

### Step 4: Move all terms to one side
Move all the terms to one side to set the equation to 0:

\[
0 = -2x^2 + 7x - 6
\]
\[
2x^2 - 7x + 6 = 0
\]

### Step 5: Solve the quadratic equation
Now, solve the quadratic equation \( 2x^2 - 7x + 6 = 0 \). We can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 2 \), \( b = -7 \), and \( c = 6 \). Plugging these values into the quadratic formula:

\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(2)(6)}}{2(2)}
\]
\[
x = \frac{7 \pm \sqrt{49 - 48}}{4}
\]
\[
x = \frac{7 \pm \sqrt{1}}{4}
\]
\[
x = \frac{7 \pm 1}{4}
\]

### Step 6: Calculate the roots
The two possible solutions are:

\[
x = \frac{7 + 1}{4} = \frac{8}{4} = 2
\]
\[
x = \frac{7 - 1}{4} = \frac{6}{4} = 1.5
\]

### Final Answer
The solutions to the equation are:

\[
x = 2 \quad \text{or} \quad x = 1.5
\]

Let me know if you'd like further details or clarification!

---

Here are 5 related questions you could try:
1. How would the equation change if the constant 3 were replaced by 5 in the numerator?
2. What happens if the linear term on the right-hand side is modified to \( 1 - 3x \)?
3. Could you solve the equation graphically by plotting both sides as functions of \( x \)?
4. How would the solution method differ if the equation involved a higher degree polynomial in \( x \)?
5. Can you verify both solutions by plugging them back into the original equation?

**Tip:** Always check for extraneous solutions when solving equations involving fractions, as multiplying by the denominator may introduce invalid solutions!

Learn how to solve the quadratic equation 3/(x - 3) = 1 - 2x with a step-by-step solution, including elimination of fractions and using the quadratic formula. This problem is suitable for students in Grades 9-11.

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