How do you factor #2x^2 + 18x + 36#?

1 Answer
Jan 17, 2017

# 2x^2+18x+36 = 2(x+3)(x+6) #

Explanation:

The rule to factorise any quadratic is to find two numbers such that

#"product" = x^2 " coefficient "xx" constant coefficient"#
#"sum" \ \ \ \ \ \ = x " coefficient"#

So for # 2x^2+18x+36 # first we note that there is a common factor of #2# which we can immediately factor out to give:

# 2x^2+18x+36 = 2(x^2+9x+18) #

we seek two numbers such that

#"product" = (1)*(18) = 18#
#"sum" \ \ \ \ \ \ = 9#

So we look at the factors of #18#. As the sum is negative and the product is positive then both factors must be positive, We can check every combination of the product factors:

# {: ("factor1", "factor2", "sum"), (1,18,19),(2,9,11),(3,6,9) :} #

So the factors we seek are #color(blue)(3)# and #color(green)(6)#. With practice one can determine the appropriate factors by inspection alone.

Therefore we can factorise the quadratic as follows:

# x^2+9x+18= x^2 color(blue)(+3)x color(green)(+6)x +18 #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= x(x+3)+6(x+3) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= (x+6)(x+3) #

Similarly if we grouped the factors the other way around we get the same answer:

# x^2+9x+18= x^2 color(green)(+6)x color(blue)(+3)x +18 #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= x(x+6)+3(x+6) #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= (x+3)(x+6) #

This approach works for all quadratics (assuming it does factorise) , The middle step in the last section can usually be skipped with practice.

Hence,

# 2x^2+18x+36 = 2(x+3)(x+6) #