Let $\alpha$ and $\beta$ be two roots of the equation $x^2 + 2x + 2 = 0$. Then,$\alpha^{15} + \beta^{15}$ is equal to:

Let $a, b, c$ be real numbers with $a \ne 0$. If $\alpha$ is a root of $a^2x^2 + bx + c = 0$,$\beta$ is a root of $a^2x^2 - bx - c = 0$,and $0 < \alpha < \beta$,then the equation $a^2x^2 + 2bx + 2c = 0$ has a root $\gamma$ that always satisfies:

$Quantity \, 1:$ Profit percent for a shopkeeper. The shopkeeper uses a faulty scale while buying cloth from a wholesaler. His scale measures $10 \%$ extra than the actual measurement. While selling,the shopkeeper gives $10 \%$ less and offers a discount of $10 \%$ to the customer.
$Quantity \, 2:$ Profit percent for a man who calculates his profit on the selling price. He marks up the cost of his item by $25 \%$ and then sells the item at a discount of $10 \%$.

Solve the given two equations and select the correct answer from the given options.
$I. \quad 3x^2 + 15x + 18 = 0$
$II. \quad 2y^2 + 15y + 27 = 0$

Roots of $ax^2 + b = 0$ are real and distinct if

Solve the given two equations and select the correct answer from the given options.
$I.$ $2x + 5y = 6$
$II.$ $5x + 11y = 9$