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To solve the quadratic equation:

\[
21x^2 - x - 2 = 0
\]

we'll use the quadratic formula:

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

where:
- \( a = 21 \)
- \( b = -1 \)
- \( c = -2 \)

### Step 1: Calculate the Discriminant
The discriminant is given by \( b^2 - 4ac \):

\[
(-1)^2 - 4 \cdot 21 \cdot (-2) = 1 + 168 = 169
\]

### Step 2: Solve for \( x \)
Since the discriminant is positive, we have two real solutions:

\[
x = \frac{-(-1) \pm \sqrt{169}}{2 \cdot 21}
\]

\[
x = \frac{1 \pm 13}{42}
\]

### Step 3: Calculate Each Solution
1. **For the positive root**:
   \[
   x = \frac{1 + 13}{42} = \frac{14}{42} = \frac{1}{3}
   \]

2. **For the negative root**:
   \[
   x = \frac{1 - 13}{42} = \frac{-12}{42} = -\frac{2}{7}
   \]

### Final Answer
The solutions are:
\[
x = \frac{1}{3}; -\frac{2}{7}
\]

Would you like more details on any part of the solution?

---

1. How do you derive the quadratic formula?
2. What does the discriminant tell us about the nature of roots?
3. How can you verify solutions to a quadratic equation?
4. Why are some quadratic equations unsolvable with real numbers?
5. How can factoring be used as an alternative to the quadratic formula?

**Tip:** The discriminant in the quadratic formula is crucial to check if real solutions exist. A positive discriminant means real solutions, zero gives a repeated real solution, and a negative discriminant indicates complex solutions.

Solve the quadratic equation 21x^2 - x - 2 = 0. If the solutions are not real, enter NA.

Find the real solutions of the quadratic equation 21x^2 - x - 2 = 0 using the quadratic formula. This example covers essential algebraic concepts, including calculating the discriminant and finding real solutions. Ideal for students in Grades 9-11.

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