To solve the quadratic equation:

$2x^2 - x - 21 = 0$

we can use the quadratic formula:

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

where $a = 2$, $b = -1$, and $c = -21$.

Step 1: Substitute the values of $a$, $b$, and $c$ into the formula

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot (-21)}}{2 \cdot 2}$

Step 2: Simplify the expression

1. Calculate $b^2$:
   $(-1)^2 = 1$

2. Calculate $4ac$:
   $4 \cdot 2 \cdot (-21) = -168$

3. Substitute back into the equation:
   $x = \frac{1 \pm \sqrt{1 + 168}}{4}$

4. Simplify under the square root:
   $x = \frac{1 \pm \sqrt{169}}{4}$

5. Calculate the square root:
   $x = \frac{1 \pm 13}{4}$

Step 3: Find the two possible values for $x$

1. First solution:
   $x = \frac{1 + 13}{4} = \frac{14}{4} = 3.5$

2. Second solution:
   $x = \frac{1 - 13}{4} = \frac{-12}{4} = -3$

Final Answer

The solutions are:

$x = 3.5 \quad \text{and} \quad x = -3$

Would you like more details on any step or have questions?

---

1. How can I solve other quadratic equations with different coefficients?
2. What are some methods to solve quadratic equations besides the quadratic formula?
3. Can the quadratic formula be used with complex solutions?
4. What does the discriminant tell us about the nature of solutions?
5. How can this solution process apply to equations in word problems?

Tip: Always check the discriminant ($b^2 - 4ac$) to determine the type and number of solutions in a quadratic equation.