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Question
Given that one root of the quadratic equation ax2 + bx + c = 0 is three times the other, show that 3b2 – 16ac.
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Solution
The given quadratic equation is
ax2 + bx + c = 0
Let α be the one root
Then other root = 3α
Now, Sum of the root = `(-b)/a`
⇒ α + 3α = `(-b)/a`
⇒ 4α = `(-b)/a`
⇒ α = `(-b)/(4a)` ...(i)
Also product of the root α x 3α
= `c/a`
= 3α2 = `c/a`
From equation (i) `3(-b/(4a))^2 = c/a`
⇒ `3 xx b^2/(16a^2) = c/a`
⇒ 3b2 = 16ac Proved.
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