### Quadratic Equation Question 5

#### Question 5 - 25 January - Shift 2

Let $a \in R$ and let $\alpha, \beta$ be the roots of the equation $x^{2}+60^{\frac{1}{4}} x+a=0$. If $\alpha^{4}+\beta^{4}=-30$, then the product of all possible values of $a$ is

## Show Answer

#### Answer: 45

#### Solution:

#### Formula: Sum and Product of Roots

$\begin{aligned} & x^2+60^{1 / 4} x+a=0 \Rightarrow \alpha+\beta=-60^{1 / 4}, \alpha \beta=a \\ & \alpha^2+\beta^2+2 \alpha \beta=\sqrt{60} \\ & \alpha^2+\beta^2=\sqrt{60}-2 a \\ & \alpha^4+\beta^4+2 \alpha^2 \beta^2=60+4 a^2-4 \sqrt{60} a \\ & \Rightarrow-30+2 a^2=60+4 a^2-4 \sqrt{60} a \\ & \Rightarrow 2 a^2-4 \sqrt{60} a+90=0 \\ & \text { Product of rots }=\frac{90}{ 2}=45 \end{aligned}$