If p and q are roots of the equation x^2 + lx + m =0, write an equation whose roots are -1/p and -1/q

Hey there!

I've been trying to solve this problem, but it looks like I'm going wrong somewhere. The answer in my book to this question is mx^2 -lx + 1=0, however, I am arriving mx^2+lx + 1 =0

I'd be grateful if someone can point out my mistake. My working is given below.

We have,

x^2 + lx + m =0

Let p and q be the roots of the equation. Then,

p = {-l + sqrt(l^2 - 4m)}/2 and q = {-l - sqrt(l^2 - 4m)}/2 ..... [Applying the quadratic formula]
=> -1/p = -2/{-l + sqrt(l^2 - 4m)} and -1/q = -2/{-l - sqrt(l^2 - 4m)}
=> -1/p = [-2{-l - sqrt(l^2 - 4m)}] / (-4m) and -1/q = -[-2{-l + sqrt(l^2 - 4m)}] / (-4m)
=> -1/p = {-l - sqrt(l^2 - 4m)}/(2m) and -1/q = {-l + sqrt(l^2 - 4m)}/ (2m) ... (i)

From (i) we know, a = m, b = l and c = 1 [where a is the coefficient of x^2, be the coefficient of x and c the the constant term]

Therefore the equation in the form ax^2 + bx + c = 0 is:

mx^2 +lx + 1 = 0

Try this:
Let p and q be the roots of x^2 + lx + m = 0.

Then, we know that p + q = -l, and pq = m.
(Why? (x - p)(x - q) = x^2 - (p+q)x + pq. Now compare coefficients.)

So, -1/p + -1/q = -(p+q)/(pq) = -(-l)/m = l/m, and (-1/p)(-1/q) = 1/(pq) = 1/m.
Hence, the desired equation is x^2 - (l/m)x + 1/m = 0, or mx^2 - lx + 1 = 0.

I hope this helps!
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P.S.: As for your computations:
p = {-l + √(l^2 - 4m)}/2 and q = {-l - √(l^2 - 4m)}/2
=> -1/p = -2/{-l + √(l^2 - 4m)} and -1/q = -2/{-l - √(l^2 - 4m)}
=> -1/p = [-2{-l - √(l^2 - 4m)}] / (-4m) and -1/q = [-2{-l + √(l^2 - 4m)}] / (-4m)

This last line should be (watching the minus signs):
-1/p = [-2{-l - √(l^2 - 4m)}] / (+4m) and -1/q = [-2{-l + √(l^2 - 4m)}] / (+4m).

Now, this yields -1/p = {l + √(l^2 - 4m)} / (2m) and -1/q = {l - √(l^2 - 4m)} / (2m).
Letting a = m, b = -l and c = 1, we are done (this is what I obtained above).