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Question
If one root of the quadratic equation ax2 + bx + c = 0 is double the other, prove that 2b2 = 9 ac.
Solution
ax2 + bx + c = 0.
Let the roots be α and 2α
Sum of roots = `(-b)/a`
⇒ α + 2α = `(-b)/a`
⇒ 3α = `(-b)/a`
⇒ α = `-(b)/(3a)` ...(i)
Product of root = `c/a`
⇒ 2α2 = `c/a`
α2 = `c/(2a), α = sqrt(c/(2a)` ...(iii)
Equation (i) = (ii)
`(-b)/(3a) = sqrt(c/(2a)) ...("squaring both side")`
`b^2/(9a^2) = c/(2a)`
⇒ 2b2 = 9ac.
Hence Proved.
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