Solving this strange quadratic
I'm stuck on this strange question:
kx^2 - (3k+2)x + (k+3) = 0 , has roots A and B.
i) Show that the roots are real and distinct for all numbers k,
ii) If A= 2/B, find K
iii) Given this value of K, without solving the equation, find the equation whose roots are -A^2 and -B^2.
I'll be honest, i've been on the first part for 2 hours now and I have no clue where to go. I've tried to use b^2 = 4ac but I can't get anywhere. Any people willing to help?
Thanks in advance!
By the way, my maths skills are only up to A Level.
kx^2 - (3k+2)x + (k+3) = 0 , has roots A and B.
i) Show that the roots are real and distinct for all numbers k,
ii) If A= 2/B, find K
iii) Given this value of K, without solving the equation, find the equation whose roots are -A^2 and -B^2.
I'll be honest, i've been on the first part for 2 hours now and I have no clue where to go. I've tried to use b^2 = 4ac but I can't get anywhere. Any people willing to help?
Thanks in advance!
By the way, my maths skills are only up to A Level.
i) For any quadratic equation, the roots are real and distinct if and only if the discriminant is positive (greater than 0).
ii) Consider the product of the roots.
iii) Substitute in the value of k from before and proceed to find the new equation.
Have you covered sum and product of roots yet?
ii) Consider the product of the roots.
iii) Substitute in the value of k from before and proceed to find the new equation.
Have you covered sum and product of roots yet?
i) you just need to compute delta and show that it is greater than 0 for any value of k( real roots <=> delta greater or equal 0 and distinct roots <=> delta it's not 0)
ii) do you know viete relations ?
iii) use viete relations again
ii) do you know viete relations ?
iii) use viete relations again
I used b^2 - 4ac, and that is 5k^2 + 4. But even without solving for K, I can tell K will be negative. But the question says f(x) has two distinct roots. I'm puzzled
Original post by BanglaBOSS
I used b^2 - 4ac, and that is 5k^2 + 4. But even without solving for K, I can tell K will be negative. But the question says f(x) has two distinct roots. I'm puzzled
If we have the discriminant to be 5k^2+4, what is the minimum value of the discriminant, and what can we deduce from this about the roots of the equation?
Original post by Integer123
If we have the discriminant to be 5k^2+4, what is the minimum value of the discriminant, and what can we deduce from this about the roots of the equation?
It means it has no real roots, but the question asks to prove that f(x) has real roots, not if it does. That's what has stumped me, as everything I do seems to disprove the question
Original post by BanglaBOSS
It means it has no real roots, but the question asks to prove that f(x) has real roots, not if it does. That's what has stumped me, as everything I do seems to disprove the question
Are you sure? If the discriminant is positive then the equation has distinct real roots, and remember that k^2 >= 0 for all k
Original post by Integer123
Are you sure? If the discriminant is positive then the equation has distinct real roots, and remember that k^2 >= 0 for all k
I'll be honest, I've tried several times and seem to end up with the same inequality mate. Can I ask a favour and ask if you could try the original question out?
Original post by BanglaBOSS
I'll be honest, I've tried several times and seem to end up with the same inequality mate. Can I ask a favour and ask if you could try the original question out?
Yeah I've done it and you have the correct expression for the discriminant, namely 5k^2+4. This means that its minimum value is 4 since k^2 >= 0 for all k and hence the equation has distinct real roots as the discriminant is always positive.
Original post by BanglaBOSS
I'll be honest, I've tried several times and seem to end up with the same inequality mate. Can I ask a favour and ask if you could try the original question out?
We've all tried it and solved it in a minute or two because it's very easy. Now we are spending time trying to help you, by leading you to the solution, step-by-step. OK, this isn't working because you lack the humility to listen but at least we tried...
You need to go back to basics and learn the fundamentals about quadratic equations as you clearly don't understand what the discriminant is telling you about the solutions. You will also need to know expressions for the sum and product of the roots of an equation later in the question.
(edited 6 years ago)
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