If #a - 2b = 15# and #ab = 11# then find the value of #a^2 + 4b^2#?

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Answer 1 · 2016-06-23T18:34:56

#a^2+4b^2=269#

#(a-2b)^2=a^2-4ab+4b^2= 15^2#

then

#a^2+4b^2=15^2+4 ab = 15^2+4 xx 11 =269#

Answer 2 · 2016-06-23T18:49:22

#a^2+4b^2=269.#

Method I

Given that, #a-2b=15#
#:. (a-2b)^2=15^2=225.#
#:. a^2-2*a*2b+4b^2=225,# i.e., #a^2-4ab+4b^2=225.#

Letting, #ab=11# in this, we have, #a^2-4(11)+4b^2=225.#

Hence, #a^2+4b^2=225+44=269.#

Method II

Notice that #(a+2b)^2+(a-2b)^2=2(a^2+4b^2).#

Here, we replace #(a+2b)^2# by #(a-2b)^2+8ab,# to get,

#(a-2b)^2+8ab+(a-2b)^2=2(a^2+4b^2),.# & now putting the given values,

#225+8*11+225=2(a^2+4b^2)=538,# so, #(a^2+4b^2)=538/2=269,# as before!