1
Question
If the three equations x2+ax+12=0, x2+bx+15=0, x2+(a+b)x+36=0 have a common possible root. Then, the sum of roots is
If the three equations x2+ax+12=0, x2+bx+15=0, x2+(a+b)x+36=0 have a common possible root. Then, the sum of roots is
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Solution
The correct options are
A 24
B −24
Let α be the common root of the three equation.
Let other roots be β,γ,δ respectively.
∴α+β=−a, α⋅β=12
α+γ=−b, α⋅γ=15
α+δ=−(a+b), α⋅δ=35
∴(α+β)+(α+γ)=−(a+b)=α+δ
∴α+β+γ=δ ⋯(1)
Also, α(β+γ+δ)=12+15+36=63
⇒α(2δ−α)=63 [from (1)]
⇒2α⋅δ−α2=63⇒72−α2=63
⇒α=±3
If α=3⇒β=4, γ=5, δ=12
∴α+β+γ+δ=24
If α=−3⇒β=−4, γ=−5, δ=−12
∴α+β+γ+δ=−24
A 24
B −24
Let α be the common root of the three equation.
Let other roots be β,γ,δ respectively.
∴α+β=−a, α⋅β=12
α+γ=−b, α⋅γ=15
α+δ=−(a+b), α⋅δ=35
∴(α+β)+(α+γ)=−(a+b)=α+δ
∴α+β+γ=δ ⋯(1)
Also, α(β+γ+δ)=12+15+36=63
⇒α(2δ−α)=63 [from (1)]
⇒2α⋅δ−α2=63⇒72−α2=63
⇒α=±3
If α=3⇒β=4, γ=5, δ=12
∴α+β+γ+δ=24
If α=−3⇒β=−4, γ=−5, δ=−12
∴α+β+γ+δ=−24
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