How do you solve n^2 + 19n + 66 = 6n2+19n+66=6 by completing the square?

2 Answers
Alan P. · Meave60
Apr 29, 2015

First simplify the process by moving the constant term to the right side of the equation;
then add whatever constant is necessary to both sides so that the left side is a square;
finally take the square root of both sides and simplify.

n^2+19n+66 = 6n2+19n+66=6

n^2+19n = -60n2+19n=−60

If n^2 + 19nn2+19n are the first two terms of a square
(n+a)^2 = (n^2+2an+a^2)(n+a)2=(n2+2an+a2)
then
a = 19/2a=192
and
a^2 = (19/2)^2 = 361/4 = 90 1/4a2=(192)2=3614=9014

n^2 + 19n + (19/2)^2 = -60 +90 1/4n2+19n+(192)2=−60+9014

(n+19/2)^2 = 30 1/4 = 121/4(n+192)2=3014=1214

n+19/2 = +-sqrt(121/4) = +- 11/2n+192=±√1214=±112

n = -19/2 +- 11/2 = (-19 +-11)/2n=−192±112=−19±112

n = -15n=−15
or
n=--4n=−−4

Meave60
Apr 29, 2015

n=-4, -15n=−4,−15

You can solve the problem by factoring.

n^2+19n+66=6n2+19n+66=6

Subtract 6 from both sides.

n^2+19n+60=0n2+19n+60=0

Factor.

4xx15=60 and 4+15=194×15=60and4+15=19

(n+4)(n+15)=0(n+4)(n+15)=0

n+4=0n+4=0

n=-4n=−4

n+5=0n+5=0

n=-5n=−5

n=-4, -15n=−4,−15

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