This is the question:

The straight line y=2p+1 intersects the curve y= x+ (p^2/x) at two distinct points. Find the range of values of p.

this is what i did:

2p +1 = x + p^2/x
simplify....
x^2 + (2p+1)x + p^2 = 0
b^2-4ac > 0
(-2p-1)^2 - 4(p^2) > 0
simplify....
4p + 1 > 0
P > -1/4

However, the correct answer should be:

p>-1/4, p is not equals to 0.

How do i get the second part "p not equals to 0"??

actually your equation should have been

x^2 - (2p+1)x + p^2 = 0
but since you are squaring b in
b^2 - 4ac > 0 it did not show up in your solution.

the restriction 'p not equal to zero' shows up when we analyse our equations

when p=0 the first equation is the horizontal line y = 1 and the second equation is no longer a curve, but rather simply y = x
these clearly intersect at ONE point (1,1), not at two.

since p = 0 falls in the domain of p > -1/4, we have to restrict it.

thanks!

To find the second condition that p is not equal to 0, you need to consider the equation y = x + (p^2/x).

When solving the quadratic equation x^2 + (2p+1)x + p^2 = 0, you obtained the condition 4p + 1 > 0, which simplifies to p > -1/4. This condition ensures that the discriminant (b^2 - 4ac) of the quadratic equation is greater than 0, indicating that the equation has two distinct solutions.

However, this condition does not account for the possibility of x being equal to 0. When x = 0, the equation y = x + (p^2/x) is undefined, leading to a division by zero error. Thus, we need the second condition p not equal to 0 to exclude this case and ensure that the curve and the line intersect at two distinct points.

To find the range of values of p, you correctly simplified the quadratic equation and obtained 4p + 1 > 0. This inequality implies that p must be greater than -1/4 in order for the equation to have real solutions. So you have correctly found the first part of the answer: p > -1/4.

Now let's consider the equation you derived: x^2 + (2p+1)x + p^2 = 0. The curved line y = x + (p^2/x) represents a hyperbolic curve. For a hyperbola, it is important to note that as x approaches 0, the value of y would tend towards infinity or negative infinity depending on the signs of 2p+1 and p^2.

If p were equal to 0, the equation would become x^2 + x = 0. This implies that x(x+1) = 0. The only solution for this equation would be x = 0. However, if x = 0, the equation for the curve y = x + (p^2/x) would be undefined, as division by zero is not defined. Therefore, for the curve and line to intersect at two distinct points, p cannot be equal to 0.

Hence, the complete answer is p > -1/4 and p is not equal to 0.