Question

# Solve the equation x4−4x2+8x+35=0 having given that one root is 2+√−3.

A
The roots of the given equation are 2±i,2±3i.
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B
The roots of the given equation are 2±i,2±3i.
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C
The roots of the given equation are ±2+i,2±3i.
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D
The roots of the given equation are ±2i,2±3i.
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Solution

## The correct option is A The roots of the given equation are −2±i,2±√3i.The equation is x4−4x2+8x+35=0. ...(1)One root of this equation is given as 2+√3i.Since the complex roots occur in conjugate pairs, the other root must be 2−√3i.∴S=4,p=7The quadratic factor corresponding to these two roots is x2−Sx+P or x2−4x+7.Then the other quadratic factor of L.H.S. of (1)is of the form x2+px+5.Hence we have the identityx4−4x2+8x+35=(x2−4x+7)(x2+px+5).Equating the coefficient of x on both sides of the above identity, we get 8=7p−20 or p=4.[Note that same value of, will be obtained by equating the coefficient of x2].Hence the other two roots of the equation are the roots of the equation x2+4x+5=0or x=−4±√16−202=−4±√−42=−2±i.Ans: A

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