Can two or more quadratic equations have the same pair of roots?

Question

Hi,

Reason for asking : Given a pair of roots can we find an unique quadratic equation?

RaghavSingla · Answer

Hi, Given a pair of roots, we can definitely find a quadratic equation. let A and B be the roots and S be the sum of the roots i.e. S = A+B; let P be the products of the roots i.e. P = AB. Then quadratic equation is: x^2 - (S)x +P = x^2 - (A+B)x + AB. Now question of whether this would be unique or not, I think it would be not be: Here's my logic: If you see graphically, quadratic equation is a parabola. Now, we can draw multiple parabolas intersecting x axis at same position for different equations of parabola. Please give kudos if this helps!

ccooley · Answer

No. If you have the roots, you have the quadratic equation. Think about it like this: if you know the roots, you can immediately calculate the equation itself, by multiplying out (x-a)(x-b) = 0. Could you create two different equations using the same pair of roots? No, because there's just one, single, clear process for going from roots -> equation.

However, technically, it depends on which quadratic equations you consider to be different. For example, the equations x^2 + 2x + 1 = 0 and 2x^2 + 4x + 2 = 0 have the exact same root. In my eyes, though, those are the 'same' quadratic equation - one of them simplifies directly into the other, so they're mathematically identical.

To RaghavSingla's comment above: try actually drawing two different parabolas using those criteria. (Remember that a parabola has to be symmetrical left-right!). You'll find that it's harder than you'd think... you can come up with parabolas that look different, but mathematically, they're actually just multiples of each other.

maha2804 · Answer

So a DS question asks us to find a quadriatic equation and one of the statement gives us the roots of the equation,is that particular statement sufficient? (Is there any OG questions of this format? I haven't gone through all the OGs yet.)

What about two parabolas who are the inverse to each other in terms of sign(x^2+2x+1 and -x^2-2x-1). These clearly have the same pair of roots but are different parabolas.

ccooley · Answer

I'd say that depends on what you mean by 'the same'. In that case, you can get from one to the other just by multiplying everything by -1. Mathematically, since one simplifies to the other, I'd say they're the same for GMAT purposes (where the point of creating an equation would probably be doing algebra with it, not drawing a parabola.)

maha2804 · Answer

My question arises due to this doubt. If in a DS question, even if there is a doubt there can two answers then we have to be careful in answering it as Sufficient. What do you suggest you do then?

ccooley · Answer

I've never seen a DS question like that. I think the reason for that is exactly what we've identified here. There's some ambiguity over what equations would be 'the same'. Using my definition, the statement would be sufficient (because there's one 'mathematically equivalent' equation/parabola that works, for any given pair of roots). Using your definition, it wouldn't (because you can create two equations/parabolas that don't look exactly alike.)

Luckily, I've never seen a DS problem that asks you 'what is the equation for suchandsuch?'. 99.9% of the time, they'll either ask you a yes/no question, or a question that can be answered using a single number. The other 0.1% of questions are ones that ask things like 'of a, b, and c, which one is the greatest?'. The closest GMAT-like example I can imagine is this one - which I just made up, so I make no promises about how good of a question it is!:

Is the value of the quadratic equation f(x) equal to 2x^2 - 10x + 12 for all values of x?

(1) f(3) = 0
(2) f(2) = 0

In that case, the correct answer would be E. Your definition is the one that 'applies' here, since you're actually asked to take values and plug them into the quadratic. Here's the proof:

(1) you have no idea what f is - insufficient.
(2) you have no idea what f is - insufficient. 
(1 + 2 together) - f(x) could equal x^2 - 5x + 6, in which case the answer would be 'no'. That's because for some values (for example, x = 0) it doesn't equal 2x^2 - 10x + 12. 
Or, f(x) could equal 2x^2 - 10x + 12, in which case the answer would be 'yes'. 
We've got both a yes and a no, so the answer is E.

Can two or more quadratic equations have the same pair of roots?

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