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Let's solve the quadratic equation \( 4w^2 - 3w - 2 = 0 \) using the given quadratic formula form:

\[
w = \frac{N \pm \sqrt{D}}{M}
\]

### (A) Step-by-Step Solution:

We will use the quadratic formula:

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

For the equation \( 4w^2 - 3w - 2 = 0 \), the coefficients are:
- \( a = 4 \)
- \( b = -3 \)
- \( c = -2 \)

#### Step 1: Compute \( D = b^2 - 4ac \)
\[
D = (-3)^2 - 4(4)(-2) = 9 + 32 = 41
\]

So, \( D = 41 \).

#### Step 2: Find \( N \) and \( M \)
From the quadratic formula, \( N = -b = 3 \) and \( M = 2a = 8 \).

Thus, the values are:
- \( N = 3 \)
- \( M = 8 \)
- \( D = 41 \)

### (B) Approximating the solutions

Now, let's calculate the approximate solutions for \( w \) using \( w = \frac{N \pm \sqrt{D}}{M} \):

\[
w = \frac{3 \pm \sqrt{41}}{8}
\]

#### Step 1: Calculate \( \sqrt{41} \)
\[
\sqrt{41} \approx 6.403
\]

#### Step 2: Calculate the two solutions
1. \( w_1 = \frac{3 + 6.403}{8} = \frac{9.403}{8} \approx 1.18 \)
2. \( w_2 = \frac{3 - 6.403}{8} = \frac{-3.403}{8} \approx -0.43 \)

Thus, the approximate solutions are:

\[
w \approx 1.18, -0.43
\]

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Would you like more details or have any questions? Here are some related questions you might find helpful:

1. What is the quadratic formula and how is it derived?
2. How do you determine if a quadratic equation has real or complex solutions?
3. Can you explain the significance of the discriminant \( D \) in quadratic equations?
4. How would the solutions change if the coefficients of the equation were different?
5. Can quadratic equations be solved using methods other than the quadratic formula?

**Tip:** Always check the discriminant (\( D \)) before solving a quadratic equation to know the nature of the roots (real or complex).

The equation 4w^2 - 3w - 2 = 0 has solutions of the form w = (N ± √D)/M. Solve this equation and find the values of N, M, and D. Approximate the solutions using a calculator and round to two decimal places.

Learn how to solve the quadratic equation 4w^2 - 3w - 2 = 0 using the quadratic formula. Find the values of N, M, and D, and calculate the approximate solutions rounded to two decimal places. This problem highlights the use of the discriminant in quadratic equations. Suitable for students in Grades 8-10.

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