Let $r$ be a root of the equation $x^2 + 2x + 6 = 0$. Then the value of the expression $(r+2)(r+3)(r+4)(r+5)$ is

Question

Twenty girls are sitting in a row. A girl is shifted by four places towards the left, then she became 8th from the left end. The earlier position of that girl from the right end of the row is

Answer 1

To find the value of the expression $(r+2)(r+3)(r+4)(r+5)$ given that $r$ is a root of the quadratic equation $x^2 + 2x + 6 = 0$, we will use properties of roots and complex numbers.

Step 1: Identify the roots of the equation.

The given quadratic equation is:

x^2 + 2x + 6 = 0

To find the roots, we use the quadratic formula:

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

where $a = 1$, $b = 2$, and $c = 6$. Substituting these values, we get:

x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}

Simplifying the expression under the square root:

\begin{align*} 2^2 - 4 \cdot 1 \cdot 6 &= 4 - 24 \\ &=-20 \end{align*} \nonumber

So, we have:

x = \frac{-2 \pm \sqrt{-20}}{2}

This simplifies to:

x = \frac{-2 \pm \sqrt{20}i}{2}

Further simplifying gives:

x = -1 \pm i\sqrt{5}

Thus, the roots of the quadratic equation are $r = -1 + i\sqrt{5}$ and the other root $r = -1 - i\sqrt{5}$.

Step 2: Evaluate the expression $(r+2)(r+3)(r+4)(r+5)$.

To evaluate this expression, substitute $r = -1 + i\sqrt{5}$ into the expression:

We use:

\[(r+2)(r+3)(r+4)(r+5)\]

r + 2 = -1 + i\sqrt{5} + 2 = 1 + i\sqrt{5}
r + 3 = -1 + i\sqrt{5} + 3 = 2 + i\sqrt{5}
r + 4 = -1 + i\sqrt{5} + 4 = 3 + i\sqrt{5}
r + 5 = -1 + i\sqrt{5} + 5 = 4 + i\sqrt{5}

So, the expression becomes:

(1 + i\sqrt{5})(2 + i\sqrt{5})(3 + i\sqrt{5})(4 + i\sqrt{5})

For simplifying, note the conjugate property of roots and properties of multiplication using conjugates:

$ (x+iy)(x-iy) = x^2 + y^2 $, this can be used to simplify calculation:

The simplified form will yield:

(1+i\sqrt{5})(4+i\sqrt{5})-(2+i\sqrt{5})(3+i\sqrt{5}) = -126

Thus, the evaluated expression is:

-126

Hence, the correct answer is: $\textbf{-126}$.

Answer 2