Question

Show that the equation \[{x^2} + 2px - 3 = 0\] has a real and distinct root for all values of p.

Answer 1

Hint: We have to check if this given equation has real and distinct roots for all values of p. So, at the beginning we will consider the discriminant term which is to be said as, \[D = {b^2} - 4ac\] for a quadratic equation \[a{x^2} + bx + c = 0\]. If that D is always greater than zero for all values of p, we will get our proof.

Complete step-by-step answer:
We know that for a quadratic equation \[a{x^2} + bx + c = 0\], discriminant \[D = {b^2} - 4ac\].
The condition for 'distinct' real roots is \[D{\text{ }} > {\text{ }}0\]. Thus, to prove that \[{x^2} + 2px - 3 = 0\] has distinct real roots, we got to prove \[D{\text{ }} > {\text{ }}0\] for it.
Let’s have a look at what D comes out for \[{x^2} + 2px - 3 = 0\]
so, we have, our,
\[D = 2{p^2} - 4(1)( - 3)\]
On simplification we get,
\[ \Rightarrow \]\[D = 4{p^2} + 12\]
We obviously understand that if p is a real number (and it's given in the question that it is a real number), \[4{p^2}\]will always be greater than or equal to 0 \[({p^2} > 0).\] The reason of it being the fact that square of any real number can never be a negative number and thus it is always greater than or equal to 0.
Now, if \[4{p^2}\]is always greater than or equal to 0, then, by common sense, we can say that \[4{p^2} + 12\] must be greater than or equal to 12. We can understand the same in the form of a mathematical inequality. We know that
\[{p^2} > 0\]
\[ \Rightarrow 4{p^2} > 0\]
By adding 12 to both sides of the inequality
\[ \Rightarrow 4{p^2} + 12 > 12\]
So, now that we know that \[4{p^2} + 12\] is greater than or equal to 12, it automatically becomes greater than 0. \[4{p^2} + 12 > 12\], which means that the least value it can take is 12. It is always going to be 12 or more than that, and thus we can say that \[4{p^2} + 12\] is also \[ > {\text{ }}0\].
Now, we have found that \[D = 4{p^2} + 12\] for the equation in the question. We've found out that \[4{p^2} + 12\] is always greater than 0.
Hence, we can say that the equation will have distinct and real roots.

Note: Here we have used the condition that the discriminant has to be greater than zero to have real and distinct roots. There might also be cases where the discriminant is only equal to zero. In that case we will get real but not distinct roots. The roots will be equal to each other.