Quadratic factoring What is a general rule of thumb that we can use to check whether a quadratic expression can be factored without the use of the quadratic formula since using the quadratic formula can be time consuming in exam situations?
Is it even possible to tell whether an expression will require application of the quadratic formula even before beginning to factor it?
 A: All second degree equations can be factored but we might ask: Are the factors real or complex? Are they rational or irrational? Another answer mentioned the discriminant.
If $\quad b^2-4ac\lt0\quad $ the factors are complex (with an imaginary parts).
If $\quad b^2-4ac\ge 0\quad $ the factors are real numbers
If $\quad b^2-4ac\qquad$ is a perfect square, the factors are rational.
If you want to "find" the roots (factors) (if they exist) without the quadratic formula, you can apply the rational root theorem. In it, you just try all of the rational numbers composed of factors of the first and last terms, dividing using synthetic division, and seeing if they divide without remainder into the original quadratic polynomial.
A: The criterion is to check whether the discriminant $$b^2-4ac$$ of the equation $$ax^2+bx+c=0$$ is a perfect square (assuming the coefficients are integers). The equation can be factorised in $\mathrm Q$ if and only if this is the case. You can quickly check whether the square root of the discriminant is an integer if you're allowed a calculator.
