To solve the quadratic equation \( 4m^2 - m - 3 = 0 \), we can use the quadratic formula:

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

Where \( a = 4 \), \( b = -1 \), and \( c = -3 \). Now, let's plug these values into the formula.

### Step 1: Calculate the discriminant
\[
\Delta = b^2 - 4ac = (-1)^2 - 4(4)(-3) = 1 + 48 = 49
\]

### Step 2: Apply the quadratic formula
\[
m = \frac{-(-1) \pm \sqrt{49}}{2(4)} = \frac{1 \pm 7}{8}
\]

### Step 3: Solve for the two possible values of \( m \)

1. \( m_1 = \frac{1 + 7}{8} = \frac{8}{8} = 1 \)
2. \( m_2 = \frac{1 - 7}{8} = \frac{-6}{8} = -\frac{3}{4} \)

### Final answer:
The solutions are \( m = 1 \) and \( m = -\frac{3}{4} \).

Would you like more details or have any questions about this solution?

Here are 5 related questions to explore:
1. What if the quadratic equation had a negative discriminant?
2. How can the quadratic formula be derived?
3. Can you solve quadratic equations by factoring?
4. What is the relationship between the discriminant and the nature of the roots?
5. How do you graph a quadratic function and interpret its solutions?

**Tip:** Always check the discriminant to see if the roots are real, imaginary, or repeated.

Explore how to solve the quadratic equation 4m² - m - 3 = 0 using the quadratic formula. This step-by-step guide demonstrates how to calculate the discriminant and find the two possible solutions. Ideal for students in Grades 8-10.

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