How do you factor by grouping #25v^2+60v+27#?

Question

Genesis Allen · Accepted Answer

#(5v+3)(5v+9)#

#color(white)(..)+color(green)(60)# #color(white)(..)xx 675# #* * * * * * # #color(white)(.)1 color(white)(0)xx 675# #color(white)(.)3 color(white)(0)xx 225# #color(white)(.)5 color(white)(0)xx 135# #color(white)(.)9 color(white)(0) xx 75# #color(white)(.)15 xx 45# #=> color(red)(15)+color(red)(45) = color(green)(60)#

So we've got

#(25v^2 + color(red)(45)v) + (color(red)(15)v + 27)#

#5v(5v+9) + 3(5v+9)#

#(5v+3)(5v+9)#

Abigail Allen · Answer

undefined To factor by grouping the expression 25v^2 + 60v + 27:

1. Multiply the leading coefficient (25) by the constant term (27):
   25 * 27 = 675.

2. Find two numbers that multiply to give 675 and add to give the coefficient of the linear term (60). These numbers are 45 and 15.

3. Rewrite the middle term (60v) using these two numbers:
   60v = 45v + 15v.

4. Group the terms:
   25v^2 + 45v + 15v + 27.

5. Factor by grouping:
   (25v^2 + 45v) + (15v + 27).

6. Factor out the greatest common factor from each group:
   5v(5v + 9) + 3(5v + 9).

7. Notice that both terms have a common factor of (5v + 9).

8. Factor out the common factor:
   (5v + 9)(5v + 3).