To factorize ax^2+bx+cax2+bx+c, one needs to first check about discriminant b^2-4acb2−4ac, which in 3x^2−15x+163x2−15x+16 is (-15)^2-4xx3xx16=225-192=33(−15)2−4×3×16=225−192=33. As it is not the square of a rational number, we will not have binomials with rational coefficients as factors.
Hence, to find factors let us solve the equation 3x^2−15x+16=03x2−15x+16=0 using quadratic formula which gives solution of ax^2+bx+c=0ax2+bx+c=0
as x=(-b+-sqrt(b^2-4ac))/(2a)x=−b±√b2−4ac2a.
Hence solution of 3x^2−15x+16=03x2−15x+16=0 is x=(-(-15)+-sqrt((-15)^2-4xx3xx16))/(2xx3)x=−(−15)±√(−15)2−4×3×162×3 or
x=(15+-sqrt33)/6x=15±√336
Hence factors are 3(x-(15+sqrt33)/6)(x-(15-sqrt33)/6)3(x−15+√336)(x−15−√336)
We have multiplied by 33 as coefficient of x^2x2 is 33.
