# Thread: Need help with Roots of Quadratic

1. ## Need help with Roots of Quadratic

5. a) Prove that the equation (k^2+l^2)x^2-4m(k-l)x+4m^2, where m does not equal to 0, and k and l are reciprocal, has no real roots.
b) Given that (k^2 -l^2 )x^2+2m(k+l)x+m^2=0 has a double root and that k, l, m does not equal to 0, prove that k=-l

2. ## Re: Need help with Roots of Quadratic

Originally Posted by wllblsstn
5. a) Prove that the equation (k^2+l^2)x^2-4m(k-l)x+4m^2, where m does not equal to 0, and k and l are reciprocal, has no real roots.
b) Given that (k^2 -l^2 )x^2+2m(k+l)x+m^2=0 has a double root and that k, l, m does not equal to 0, prove that k=-l
Have you tried using the discriminant? What do you know about what the discriminant looks like based on whether the quadratic has no real roots, or has a double root?

3. ## Re: Need help with Roots of Quadratic

Originally Posted by InteGrand
Have you tried using the discriminant? What do you know about what the discriminant looks like based on whether the quadratic has no real roots, or has a double root?
Hmm yes I do realise that b^2-4ac needs to be less than 0 in order for it to have no real roots, and equal to zero in order for it to have double roots, but I believe that step comes after finding 'k' and 'l' as it does mention that they are reciprocal, thus k=1/land yeah.

4. ## Re: Need help with Roots of Quadratic

Originally Posted by wllblsstn
Hmm yes I do realise that b^2-4ac needs to be less than 0 in order for it to have no real roots, and equal to zero in order for it to have double roots, but I believe that step comes after finding 'k' and 'l' as it does mention that they are reciprocal, thus k=1/land yeah.
You only need the substitution $l = 1/ k$ and it is possible to prove the resulting expression is always negative regardless of the value of $k$.

5. ## Re: Need help with Roots of Quadratic

Originally Posted by fan96
You only need the substitution $l = 1/ k$ and it is possible to prove the resulting expression is always negative regardless of the value of $k$.
aah how did I not think of this. Thank you, but are you still able to provide more help, since I still don't see how variables 'm' and 'l' could prove that a discriminant is less than 0. Thanks

6. ## Re: Need help with Roots of Quadratic

Originally Posted by wllblsstn
aah how did I not think of this. Thank you, but are you still able to provide more help, since I still don't see how variables 'm' and 'l' could prove that a discriminant is less than 0. Thanks
Let $l = 1/k$.

\begin{aligned} \Delta &= (4m)^2\left(k-\frac 1k\right)^2 - 4(4m^2)\left(k^2 + \frac{1}{k^2}\right) \\ &= 16m^2\left(k^2 - 2 + \frac{1}{k^2} - k^2 - \frac{1}{k^2}\right) \\ &= 16m^2(-2) \\ &= -32m^2\end{aligned}

which is negative for all nonzero real $m$.

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