To solve the quadratic equation \(3x^2 - 2x = 1\), follow these steps:

1. **Move all terms to one side** to set the equation equal to zero:
   \[
   3x^2 - 2x - 1 = 0
   \]

2. **Factor the quadratic equation** (if possible). If it can't be easily factored, use the quadratic formula:
   The quadratic formula is given by:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   For the equation \(3x^2 - 2x - 1 = 0\), the coefficients are:
   \[
   a = 3, \quad b = -2, \quad c = -1
   \]

3. **Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula**:
   \[
   x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(-1)}}{2(3)}
   \]
   Simplify the equation step by step:
   \[
   x = \frac{2 \pm \sqrt{4 + 12}}{6}
   \]
   \[
   x = \frac{2 \pm \sqrt{16}}{6}
   \]
   \[
   x = \frac{2 \pm 4}{6}
   \]

4. **Solve for the two possible values of \(x\)**:
   \[
   x = \frac{2 + 4}{6} = \frac{6}{6} = 1
   \]
   \[
   x = \frac{2 - 4}{6} = \frac{-2}{6} = -\frac{1}{3}
   \]

The solutions are \(x = 1\) and \(x = -\frac{1}{3}\).

Would you like any further details or explanations?

Here are some related questions:

1. How would the solutions change if the equation was \(3x^2 - 2x + 1 = 0\)?
2. What is the discriminant, and how does it relate to the number of real solutions?
3. How can this quadratic equation be solved by completing the square?
4. What do the solutions represent graphically on a parabola?
5. How would you solve \(3x^2 - 2x = -1\)?

**Tip:** The discriminant (\(b^2 - 4ac\)) tells you how many real solutions a quadratic equation has.

Learn how to solve the quadratic equation 3x^2 - 2x = 1 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula and factoring quadratic equations. Suitable for students in Grades 10-12.

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