Question

Find the value of a/b+b/a, if a and b are the roots of the quadratic equation x^2+8x+4=0

Answer 1

GIVEN :- P(x) = x^2 + 8x + 4 = 0

TO FIND :- Value of

a/b + b/a where, a and b are zeros of the polynomial.

SOLUTION :-

We know that the

$sum \: of \: zeros = - \frac{coefficient \: of \: {x}^{} }{coefficient \: of \: {x}^{2} } \\ \\ \\ \\ \implies \boxed{a + b = - \frac{8}{1} = - 8}$

Also we know that ,

$product \: of \: the \: zeros = \frac{constant}{coefficient \: of \: {x}^{2} } \\ \\ \\ \\ \implies product \: of \: the \: zeros = \frac{4}{1} \\ \\ \\ \implies \boxed{ab = 4}$

Now , we will be finding value of the following :-

$\frac{a}{b} + \frac{b}{a} \\ \\ \\ \\ = \frac{ {a}^{2} + {b}^{2} }{ab} \\ \\ \\ \\ = \frac{ {(a + b)}^{2} - 2ab }{ab} \\ \\ \\ \\ = \frac{ { (- 8)}^{2} - 2 \times 4 }{4} \\ \\ \\ \\ = \frac{64 - 8}{4} \\ \\ \\ \\ = \frac{56}{4} \\ \\ \\ = \boxed{ \boxed{14}}$

Hence the Answer is 14

N.B.

While solving such problems we should keep in mind that we have to first simplify the given equation to a + b and ab so that we can substitute that value.

e.g. In this case we reduce

(a^2 + b^2) in to ( a + b )^2 - 2ab