If one root of equation x2+px+q=0 is √3+2 then the value of (p,q) is -
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Answer :. please give me thanks
Given that [2 + i.sqrt(3)] is a root of x^2 + px + q = 0
We know that if a quadratic equation has non-real (imaginary) roots, its roots always exist in conjugate pairs. for eg., (a + ib) and (a - ib)
Also note that i = -sqrt(1) and because it doesn't really exists or it can't be located on the real number line, its said to be a non real or an imaginary or a complex number. This forms a completely new branch called Complex Numbers, whose questions and basics will be discussed later.
According to the rule of conjugate pairs, the roots of x^2 + px + q = 0 are
[2 + i.sqrt(3)] and [2 - i.sqrt(3)]
Now, for x^2 + px + q = 0;
sum or roots = -p
=> [2 + i.sqrt(3)] + [2 - i.sqrt(3)] = -p
=> 4 = -p
=> p = -4
Similarly,
Product of roots = q
=> [2 + i.sqrt(3)]*[2 - i.sqrt(3)] = q
=> (2)^2 - (i.sqrt(3))^2 = q (using (a+b)(a-b) = a^2 - b^2)
=> 4 - i^2 3= q
=> 4 - (-3) = q
=> q = 7
So, p= -4 and q = 7
Step-by-step explanation:
by substituting √3+2 in place of x and simplifying it we get values of p and q