How do you write two binomial in the form #asqrtb+csqrtf# and #asqrtb-csqrtf#?
1 Answer
An example would be:
#2sqrt(3)+5sqrt(7)" "# and#" "2sqrt(3)-5sqrt(7)#
Explanation:
I think you just did.
The expressions:
#asqrt(b)+csqrt(f)" "# and#" "asqrt(b)-csqrt(f)#
are already binomials, so it would seem that the answer is in the question.
I am not sure what is really wanted, except that we could substitute numerical values for the variables.
For example, with:
#{(a=2),(b=3),(c=5),(f=7):}#
we have:
#2sqrt(3)+5sqrt(7)" "# and#" "2sqrt(3)-5sqrt(7)#
What is interesting about these expressions is that they are radical conjugates of one another. If you mutiply the two binomials together you will get a rational result (assuming the coefficients are rational).
In general, we find:
#(asqrt(b)+csqrt(f))(asqrt(b)-csqrt(f)) = (asqrt(b))^2-(csqrt(f))^2#
#color(white)((asqrt(b)+csqrt(f))(asqrt(b)-csqrt(f))) = a^2b-c^2f#
and with our choice of coefficients we find:
#(2sqrt(3)+5sqrt(7))(2sqrt(3)-5sqrt(7)) = 2^2(3)-5^2(7) = 12-175 = -163#
