1. ### Ak94

6
1. The problem statement, all variables and given/known data
What is the condition, for a quadratic function of the form
ax2 + bx + c = y
to be a perfect square? (x, y are real here)

There's a question of this type in a book I'm working with, and I'd just like to have some general conditions for any quadratic...

3. The attempt at a solution

Since y = ax2 + bx + c = a(x-$$\alpha$$)(x-$$\beta$$)
where $$\alpha$$ and $$\beta$$ are the values of x for which y = 0,
y is a perfect square when Discriminant of quadratic = 0 (this ensures that $$\alpha$$ = $$\beta$$) and when a is a perect square..

Are these the required conditions for any quadratic function (of the given form) to be a perfect square? Any condition I may have missed?

### Staff: Mentor

How about if y = 2x2 + 4x + 2 = 2(x + 1)(x + 1)? Is that a perfect square?

3. ### Ak94

6
hm..
2x2+4x+2=[$$\sqrt{2}$$(x+1)]2
and since $$\sqrt{2}$$ isn't an integer, the expression isn't a perfect square...
so that does imply that when D = 0, and coefficient of x2 is a perfect square, we can conclude that the entire quadratic will be one.. ?

4. ### HallsofIvy

40,943
Staff Emeritus
So I guess it depends on what your definition of "perfect square" is! And I've never seen one that required constants to be integers. Is that from your textbook?

1,478
Are you required to use integers as coefficients? I know some introductory algebra books specify that you factor only if you have integer coefficients, but you have not specified this. If so, your comment that

$$2x^2 + 4x + 2 = \left( \sqrt 2 (x+1) \right)^2$$

is not a perfect square is correct. If non-integer coefficients are allowed, it is a perfect
square.

6. ### Ak94

6
it's kind of ambigously mentioned in the book, so I'll talk to my teacher about it.., thanks for pointing that out :)

So a perfect square number would have to be the square of an integer, but that is not needed for an expression.. ?

Also.. I guess D = 0 is still a necessary condition
Since this perfect square would be greater than or equal to 0, the parabola representing the function would have to open upwards, so it would be greater than 0

So, the necessary (and sufficient) conditions are D = 0, a > 0? (assuming that any real coefficient is allowed)
Is that right?

7. ### HallsofIvy

40,943
Staff Emeritus
Not necessarily. I would call "$a(x- b)^2$" a "perfect square" for any numbers a and b- a does not have to be positive.

8. ### Ak94

6
I see.. so D = 0 is the only condition..
to everyone who helped out, thanks :)