How would you factorise an equation with no factors?

Question

So I need to factorise the equation: x²-5x-7=0

I have learned how to do this before, but can’t for the life of me remember or find the theory for it.

Thanks if you can lend me a hand!

Answer 1

If there's an integer factor, you'd look at the constant term, 7, and consider its factors, 7 and 1. But 7-1 = 6 not 5, so there are no integer factors Had the problem been x^2 -6x -7, you could have factored it into (x-7)(x+1).

use the quadratic formula: x = -b/2a + or - (1/2a)sqr(b^2-4ac) where a=1, the coefficient of the x^2 term, b=-5 the coefficient of the x term, and c=-7 the constant term

then x=5/2 + or - (1/2)sqr(25-4(1)(-7)) = 2.5 + or - (1/2(sqr53)= 2.5 + (sqr53)/2 or 2.5 - (sqr53)/2

then the non-integer irrational factors are (x-2.5-sqr53/2)(x-2.5+sqr53/2) = 0

that's the only way to factor the given equation, into irrational factors. No one ever really does that though. They just look for integer factors, of which there are none in this case.