It seems like there might be a small typo in the equation you provided. Based on the context, I assume you meant to write the quadratic equation as:

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

Let me solve it for you.

### Step 1: Move all terms to one side of the equation
Subtract 3 from both sides to set the equation to zero:

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

### Step 2: Apply the quadratic formula
The quadratic formula is:

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

For the equation \(2x^2 - x - 3 = 0\), the coefficients are:
- \(a = 2\)
- \(b = -1\)
- \(c = -3\)

### Step 3: Calculate the discriminant
\[
\text{Discriminant} = b^2 - 4ac = (-1)^2 - 4(2)(-3) = 1 + 24 = 25
\]

### Step 4: Calculate the roots
\[
x = \frac{-(-1) \pm \sqrt{25}}{2(2)} = \frac{1 \pm 5}{4}
\]

This gives us two possible solutions:
1. \(x = \frac{1 + 5}{4} = \frac{6}{4} = 1.5\)
2. \(x = \frac{1 - 5}{4} = \frac{-4}{4} = -1\)

### Final Answer:
The solutions to the quadratic equation \(2x^2 - x - 3 = 0\) are:

\[
x = 1.5 \quad \text{and} \quad x = -1
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. How can you determine the nature of the roots of a quadratic equation using the discriminant?
2. What are the properties of the graph of the quadratic equation \(y = 2x^2 - x - 3\)?
3. How do you factorize a quadratic equation, and can this equation be factorized?
4. What is the significance of the vertex in the graph of a quadratic equation?
5. How would you solve a quadratic equation by completing the square?

**Tip:** Always check your equation for possible typographical errors before solving, especially in math problems!

Learn how to solve the quadratic equation 2x^2 - x - 3 with a detailed step-by-step solution. This problem covers quadratic equations, discriminant calculation, and the application of the quadratic formula. Suitable for students in Grades 10-12.

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