Find the quadratic equation whose roots $\frac{2}{3} are -\frac{1}{2}$ and :

Question

Accepted Answer

To determine the correct quadratic equation whose roots are $\frac{2}{3}$ and $-\frac{1}{2}$, we need to use the properties of quadratic equations and the relationships between the roots and coefficients.

### Explanation for Option 2:

The general form of a quadratic equation is $ax^2 + bx + c = 0$. For a quadratic equation with roots $p$ and $q$, the sum and product of the roots are given by:

1. Sum of the roots, $p + q = -\frac{b}{a}$
2. Product of the roots, $pq = \frac{c}{a}$

Given roots are $\frac{2}{3}$ and $-\frac{1}{2}$.

#### Step 1: Calculate the sum of the roots
$$
\frac{2}{3} + \left(-\frac{1}{2}\right) = \frac{2}{3} - \frac{1}{2} = \frac{4}{6} - \frac{3}{6} = \frac{1}{6}
$$

According to the sum of roots formula:
$$
-\frac{b}{a} = \frac{1}{6}
$$

#### Step 2: Calculate the product of the roots
$$
\frac{2}{3} \times \left(-\frac{1}{2}\right) = -\frac{2}{6} = -\frac{1}{3}
$$

According to the product of roots formula:
$$
\frac{c}{a} = -\frac{1}{3}
$$

Now, let's verify Option 2: $6x^2 - x - 2 = 0$.

- For this equation, $a = 6$, $b = -1$, and $c = -2$.

#### Check the sum of the roots:
$$
-\frac{b}{a} = -\left(-\frac{1}{6}\right) = \frac{1}{6}
$$

This matches the calculated sum of the roots.

#### Check the product of the roots:
$$
\frac{c}{a} = \frac{-2}{6} = -\frac{1}{3}
$$

This matches the calculated product of the roots.

Thus, Option 2 satisfies both the sum and product of the roots, confirming it as the correct quadratic equation.

### Explanation for why other options are incorrect:

- **Option 1: $6x^2 + x + 2 = 0$**
  - Sum of roots: $-\frac{1}{6}$ (does not match $\frac{1}{6}$)
  - Product of roots: $\frac{1}{3}$ (does not match $-\frac{1}{3}$)

- **Option 3: $16x^2 - x + 2 = 0$**
  - Sum of roots: $\frac{1}{16}$ (does not match $\frac{1}{6}$)
  - Product of roots: $\frac{1}{8}$ (does not match $-\frac{1}{3}$)

- **Option 4: $16x^2 + x - 2 = 0$**
  - Sum of roots: $-\frac{1}{16}$ (does not match $\frac{1}{6}$)
  - Product of roots: $-\frac{1}{8}$ (does not match $-\frac{1}{3}$)

Each of these options fails to satisfy either the sum or the product of the roots, confirming that they are incorrect. Therefore, Option 2 is the only correct choice.