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  4. If a>2, then the roots of the equation (2 - a)x2+3ax-1=0 are

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Q. If $a>2,$ then the roots of the equation $\left(2 - a\right)x^{2}+3ax-1=0$ are

NTA AbhyasNTA Abhyas 2020Complex Numbers and Quadratic Equations

Solution:

Let, $f\left(x\right)=x^{2}-\frac{3 a}{a - 2}x+\frac{1}{a - 2}=0$
$D=\frac{9 a^{2}}{\left(a - 2\right)^{2}}-4\frac{1}{\left(a - 2\right)}=\frac{9 a^{2} - 4 a + 8}{\left(a - 2\right)^{2}}$
$=\frac{8 a^{2} + \left(a - 2\right)^{2} + 4}{\left(a - 2\right)^{2}}>0$
$f\left(0\right)=\frac{1}{a - 2}>0$ and
$\frac{- B}{2 A}=\frac{3 a}{2 \left(a - 2\right)}>0$
Since, $D>0,f\left(0\right)>0,\frac{- B}{2 A}>0$
Hence, both roots of the given equation are positive.