universehk wrote:

For a quadratic function f(x)=ax^2 + bx + c, we may be aware that, in reality, f(x) can always be greater than zero. In such case, there will be no solution for x.

So far I have not encountered such cases when I am having my revision in GMAT Quantitative session. BUT, will such cases happen in GMAT?

Thanks a lot1

Understand what is meant by f(x) = ax^2 + bx + c

If we want to depict this equation on the coordinate axis, we say

y = ax^2 + bx + c is an upward sloping parabola if a is positive.

What is the meaning of solving ax^2 + bx + c = 0 for x? It means, when y = 0, what is the value of x? So you are looking for x intercepts.

What is the meaning of solving ax^2 + bx + c = d for x? It means, when y = d, what is the value of x? Depending on the values of a, b, c and d, you may or may not get values for x.

e.g. x^2 - 2x - 3 = 0

(x + 1)(x - 3) = 0

x = -1 or 3

This is what it looks like:

Attachment:

images.jpeg [ 6.03 KiB | Viewed 897 times ]
So what do you do when you have x^2 - 2x -3 = -3?

You solve it in the same way:

x^2 - 2x -3 + 3 = 0

x(x - 2) = 0

x = 0 or 2

So when y is -3, x is 0 or 2.

Similarly, you can solve for it when y = 5 and get two values for x.

What happens when you put y = -5?

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