How do you factor #4x^2-15x+9#?

When you want to find the factors of a quadratic trinomial, there is a lot of information in the quadratic itself.

#ax^2 +bx + c# describes a quadratic trinomial.

#4x^2 -15x+9#

Find factors of #a# and #c#
Find factors of #4 and 9# which combine and ADD [because of PLUS 9.] to give 15

The signs in the brackets will be the SAME (because of PLUS 9).
they will BOTH be NEGATIVE (because of negative 15.)

#" "4 " and "9" "larr# find factors and cross multiply
#" "darr" "darr#
#" "4 " " 3 rarr 1 xx 3 = 3#
#" "1 " " 3 rarr 4 xx 3 = ul12#
#color(white)(....................... ...................)15" "larr 3+12#

The top row gives the one bracket, the second row gives the other bracket.

This gives #(4x-3)(1x-3)#

Multiplying out the brackets will give the original trinomial.

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Note: There is a clue in the fact that 15 is odd. An odd number is formed from an odd + even. This tells us that 4 is not split as 2 x 2, because then both products would be even. Even + even = even

Therefore 4 has to be used as #4xx1#
Once you know that there are only 3 possibilities to try for 9:

#9 xx 1, 1 xx 9 and 3 xx3#