4y^2+22y+10

Question

what is the perfect square trinomial?

Answer 1

This can be written 2(2y²+11y+5)=2(2y+1)(y+5).

When y=1, this is 2(3)(6)=6². [2(2y+1)=y+5, 4y+2=y+5, 3y=3, y=1 to make two equal factors]

When y=-5 or -½, it’s 0=0².

So the expression is a perfect square when y=-5, -½ or 1.

By solving 4(2y+1)=(y+5)/2 (which gives the same product as 2(2y+1)(y+5)) we get 16y+8=y+5, 15y=-3, y=-⅕, we get 2(⅗)(24/5)=144/25=(12/5)². There is a whole series of values which will yield perfect squares for the expression: y=(5-2n²)/(4n²-1) where n is any number≠½, yielding (18n/(4n²-1))² as the perfect square.