The quadratic equation provided is:

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

To solve this quadratic equation, we can use the quadratic formula:

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

For this equation:

• $a = 21$
• $b = -1$
• $c = -2$

Now, let’s calculate the discriminant ($\Delta$) first, which is $b^2 - 4ac$:

$\Delta = (-1)^2 - 4(21)(-2)$

Let me calculate that.The discriminant ($\Delta$) is 169, which is a positive perfect square. This means that the quadratic equation has two real and rational solutions.

Now, let's compute the solutions using the quadratic formula:

$x = \frac{-(-1) \pm \sqrt{169}}{2(21)}$

Let me calculate the two possible values for $x$.The two solutions to the quadratic equation $21x^2 - x - 2 = 0$ are:

$x_1 = 0.3333 \quad \text{(or} \quad \frac{1}{3})$ $x_2 = -0.2857 \quad \text{(or approximately} \quad -\frac{2}{7})$

Would you like further details on how these were derived or help with entering these values?

Here are 5 related questions you might explore:

1. How do you derive the quadratic formula?
2. What does the discriminant tell us about the nature of the solutions?
3. How do we know when a quadratic has irrational roots?
4. Can every quadratic equation be solved using factoring instead of the quadratic formula?
5. What are the real-life applications of solving quadratic equations?

Tip: Always check the discriminant before solving to know whether you'll have real or complex solutions.