Quadratic Equations

[kalyan601]

The equation x² + kx + 3 = 0, where k is a constant, has no real roots.
Find the set of possible values of k, giving your answer in surd form.

So the part of the quadratic formulat √b²-4ac has to be negative or = to 0 to have no real roots
a = 1
b = k
c = 3
so √k²-12 has to be negative
so k has to be <= 3 (if k is an integer which I presume it is) for this to work.
So what does it mean by giving the answer in surd form. Do I just write k<[FONT=Lucida Grande, tahoma, verdana, arial, sans-serif]√12 as my answer?[/FONT]

 

[stapel]

The equation x² + kx + 3 = 0, where k is a constant, has no real roots.
Find the set of possible values of k, giving your answer in surd form.

So the part of the quadratic formulat √b²-4ac has to be negative or = to 0 to have no real roots

Almost. The "discriminant" (being the part that's inside the "sqrt[b^2 - 4ac]") must be negative. If it's zero, then there is one real root that is repeated, like for (x + 4)^2 = x^2 + 8x + 16. (Thank you, by the way, for writing out your reasoning so nicely!)

a = 1
b = k
c = 3
so

√k²-12 has to be negative
so k has to be <= 3 (if k is an integer which I presume it is) for this to work.

Actually, I see no reason (from the posted exercise) to assume that "k" must be an integer. In fact, since they say "possible values of k, giving your answer in surd [square-root] form", it would appear that they explicitly do not want only integers for your solution.

So re-solve the inequality, but this time do it assuming that square roots are allowed in the solution.

 

[Ishuda]

...Do I just write k<√12 as my answer?

Almost. Surd form, as I understand it, is the simplified exact form. What that means is you take all integer squares out of the number inside the square root and bring it outside the square root. Take the √12 for example:
12 = 4 * 3 = 2² * 3
so
√12 = 2 √3
and
k < 2 √3
is in surd form.

 

[Quaid]

So the part of the quadratic formula √(b²-4ac) has to be negative

With any quadratic polynomial, the radical above will never be negative. A radical always denotes a non-negative Real number OR a Complex number with an imaginary part.

It is the expression b^2 - 4ac that has to be negative, not the radical itself.

The expression b^2 - 4ac is called the Discriminant.

When the Discriminant is negative, the radical √(b^2 - 4ac) does not represent a Real number.

 

[kalyan601]

With any quadratic polynomial, the radical above will never be negative. A radical always denotes a non-negative Real number OR a Complex number with an imaginary part.

It is the expression b^2 - 4ac that has to be negative, not the radical itself.

The expression b^2 - 4ac is called the Discriminant.

When the Discriminant is negative, the radical √(b^2 - 4ac) does not represent a Real number.

Almost. Surd form, as I understand it, is the simplified exact form. What that means is you take all integer squares out of the number inside the square root and bring it outside the square root. Take the √12 for example:
12 = 4 * 3 = 2² * 3
so
√12 = 2 √3
and
k < 2 √3
is in surd form.

Almost. The "discriminant" (being the part that's inside the "sqrt[b^2 - 4ac]") must be negative. If it's zero, then there is one real root that is repeated, like for (x + 4)^2 = x^2 + 8x + 16. (Thank you, by the way, for writing out your reasoning so nicely!)


Actually, I see no reason (from the posted exercise) to assume that "k" must be an integer. In fact, since they say "possible values of k, giving your answer in surd [square-root] form", it would appear that they explicitly do not want only integers for your solution.

So re-solve the inequality, but this time do it assuming that square roots are allowed in the solution.

Thanks everyone for your help!