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!