# Can all quadratic equations be factorised?

Kyoma

## Homework Statement

Is it true that all quadratic equations (ax2+bx+c) can be factorised into this form: (ax+b)(cx+d)?

I think so, but I'm not certain.

Homework Helper

## Homework Statement

Is it true that all quadratic equations (ax2+bx+c) can be factorised into this form: (ax+b)(cx+d)?

I think so, but I'm not certain.

If d is just some arbitrary number and you didn't accidentally duplicate your letters, then no.

Example:

y = x2-2x+1

So a = 1, b = -2, c = 1

So y = (x-1)(x-1)

a, c and d are fine, but you have a -1 instead of b.

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Homework Helper
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## Homework Statement

Is it true that all quadratic equations (ax2+bx+c) can be factorised into this form: (ax+b)(cx+d)?

I think so, but I'm not certain.
As gb7nash said the coefficients a, b, and c, in (ax2+bx+c) are not, in general, the same as those in (ax+b)(cx+d).

So, can (ax2+bx+c) always be factored into the form, (px+q)(rx+s)?
If you restrict q and s to be real numbers, then you must have $\sqrt{b^2-4ac}\ge0$
If q and s must be rational numbers, then you must have $\sqrt{b^2-4ac}$ be a perfect square.​