The equation whose roots are twice the roots of the equation $x^2 – 2x + 4 = 0$ is :

Question

Accepted Answer

To determine the equation whose roots are twice the roots of the given equation $x^2 - 2x + 4 = 0$, we need to follow these steps:

1. **Find the roots of the original equation:**

The given equation is $x^2 - 2x + 4 = 0$. To find its roots, we use the quadratic formula:
   $$
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   $$
   where $a = 1$, $b = -2$, and $c = 4$.

$$
   x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}
   $$
   $$
   x = \frac{2 \pm \sqrt{4 - 16}}{2}
   $$
   $$
   x = \frac{2 \pm \sqrt{-12}}{2}
   $$
   $$
   x = \frac{2 \pm 2i\sqrt{3}}{2}
   $$
   $$
   x = 1 \pm i\sqrt{3}
   $$

The roots of the equation are $1 + i\sqrt{3}$ and $1 - i\sqrt{3}$.

2. **Determine the roots of the new equation:**

We need an equation whose roots are twice these roots. Therefore, the new roots are:
   $$
   2(1 + i\sqrt{3}) = 2 + 2i\sqrt{3}
   $$
   $$
   2(1 - i\sqrt{3}) = 2 - 2i\sqrt{3}
   $$

3. **Form the new quadratic equation:**

If the roots of a quadratic equation are $p$ and $q$, the equation can be written as:
   $$
   x^2 - (p+q)x + pq = 0
   $$

Here, $p = 2 + 2i\sqrt{3}$ and $q = 2 - 2i\sqrt{3}$.

Calculate $p + q$:
   $$
   p + q = (2 + 2i\sqrt{3}) + (2 - 2i\sqrt{3}) = 4
   $$

Calculate $pq$:
   $$
   pq = (2 + 2i\sqrt{3})(2 - 2i\sqrt{3}) = 4 - (2i\sqrt{3})^2 = 4 - 4(-3) = 4 + 12 = 16
   $$

Therefore, the new equation is:
   $$
   x^2 - 4x + 16 = 0
   $$

4. **Justification for Option 4:**

The equation $x^2 - 4x + 16 = 0$ matches the derived equation with roots that are twice the roots of the original equation. Therefore, Option 4 is correct.

5. **Explanation of why other options are incorrect:**

- **Option 1: $x^2 - 2x + 4 = 0$** is the original equation and does not have roots that are twice the original roots.
   - **Option 2: $x^2 - 2x + 16 = 0$** has the wrong coefficients for both $x$ and the constant term, not matching the derived equation.
   - **Option 3: $x^2 - 4x + 8 = 0$** has the correct coefficient for $x$ but an incorrect constant term, not matching the derived equation.

Thus, the correct answer is Option 4: $x^2 - 4x + 16 = 0$.