Question

3. The condition that root of ax? + bx + c = 0 may be the double the
other is​

Answer 1

Let us take p(x) = ax²+bx+c = 0  Let α and β be the roots of p(x)  

According to the given condition,

α = 2β  

We know α+β = $\frac{-b}{a}$

i.e., 3β=$\frac{-b}{a}$ ....... (1)

αβ = $\frac{c}{a}$  

2β²= $\frac{c}{a}$ ......(2)

 Let us take p(x) = ax²+bx+c = 0            

       

Dividing by a throughout  

x²-$\frac{-b}{a}$x+$\frac{c}{a}$ = 0  

From (1) and (2)  p(x) = x²-3βx+2β² = 0

If the quadratic equation can be expressed in the above form, one of the roots will be double of the other  

Hope it helps

Answer 2

Answer:

$2b^{2}= 9ac$

Step-by-step explanation:

$\beta = 2\alpha$

using this lets solve

$\alpha +2\alpha =3\alpha = \frac{-b}{a}\\\alpha = \frac{-b}{3a}$

that is sum of the roots now for the product of the roots

$\alpha \times 2\alpha = 2\alpha ^{2}= c/a\\\alpha ^{2} = \frac{c}{2a} \\\alpha = \sqrt{\frac{c}{2a} }$

now $\alpha =\alpha$

so let solve

$\frac{-b}{3a} = \sqrt{\frac{c}{2a} }$

squaring on both sides

$\frac{b^{2} }{3a^{2} } = \frac{c}{2a}$

now cross multiplying we get

$2ab^{2}= 9a^{2}c$

now when we divide both sides by a we get

$2b^{2} = 9ac$