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  4. Consider the equation log 2(x2-5 x+5) ⋅ log 5( log 2((x+3) x+4))=0 whose roots are α, β, γ and δ, where α<β<γ<δ. The quadratic equation whose roots are α+β and γ+δ is

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Q. Consider the equation $\log _2\left(x^2-5 x+5\right) \cdot \log _5\left(\log _2((x+3) x+4)\right)=0$ whose roots are $\alpha, \beta, \gamma$ and $\delta$, where $\alpha<\beta<\gamma<\delta$.
The quadratic equation whose roots are $\alpha+\beta$ and $\gamma+\delta$ is

Complex Numbers and Quadratic Equations

Solution:

$\log _2\left(x^2-5 x+5\right) \cdot \log _5\left(\log _2((x+3) x+4)\right)=0 $
$\log _2\left(x^2-5 x+5\right)=0 $ OR $ \log _5\left(\log _2((x+3) x+4)\right)=0$
$\Rightarrow x^2-5 x+5=1 \Rightarrow \log _2((x+3) x+4)=1$
$\Rightarrow x^2-5 x+4=0 \Rightarrow x^2+3 x+4=2$
$\Rightarrow x=1,4 \Rightarrow x^2+3 x+2=0 $
$ \Rightarrow x=-1,-2 $
$\therefore \alpha=-2, \beta=-1, \gamma=1, \delta=4$
$\alpha+\beta=-3, \gamma+\delta=5$
$\therefore$ quadratic equation whose roots are $\alpha+\beta$ and $\gamma+\delta$, is
$x^2-2 x-15=0$