To solve the quadratic equation \( 2x^2 - x - 3 = 0 \) using the zero-factor property, follow these steps:

### Step 1: Factor the quadratic expression
We are looking for two numbers that multiply to give the product of the first coefficient (2) and the constant term (-3), which is \( 2 \times -3 = -6 \), and add to give the middle coefficient (-1).

The two numbers that work are -3 and 2 because:

\[
-3 \times 2 = -6 \quad \text{and} \quad -3 + 2 = -1
\]

Now, split the middle term (-x) using these two numbers:

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

### Step 2: Factor by grouping
Group the terms into pairs:

\[
(2x^2 - 3x) + (2x - 3) = 0
\]

Factor out the greatest common factor from each group:

\[
x(2x - 3) + 1(2x - 3) = 0
\]

### Step 3: Apply the zero-factor property
Now, factor out the common binomial factor \( (2x - 3) \):

\[
(2x - 3)(x + 1) = 0
\]

### Step 4: Solve each factor for zero
Set each factor equal to zero:

1. \( 2x - 3 = 0 \)
   \[
   2x = 3 \quad \Rightarrow \quad x = \frac{3}{2}
   \]
   
2. \( x + 1 = 0 \)
   \[
   x = -1
   \]

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

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

Would you like further details or have any questions about this?

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Here are 5 related questions to explore:
1. How can you check the solutions of a quadratic equation?
2. What if a quadratic cannot be factored? How would you solve it?
3. What is the difference between factoring and completing the square?
4. How would the solution change if the equation was \(2x^2 - x + 3 = 0\)?
5. How does the quadratic formula relate to the zero-factor property?

**Tip:** Always verify your factored expression by multiplying the factors back out to ensure they equal the original equation.

2x^2 - x - 3 = 0 using zero-factor property

Learn how to solve the quadratic equation 2x^2 - x - 3 = 0 step-by-step using the zero-factor property. This guide explains key algebraic concepts like factoring quadratic equations and applying the zero-factor theorem. Perfect for students in Grades 8-10.

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