36n^2+102n−84

generals336

generals336

Answered question

2020-10-21

36n2+102n84

Answer & Explanation

yunitsiL

yunitsiL

Skilled2020-10-22Added 108 answers

To factor 36n2+102n84, we first need to factor out the GCF of the terms.

Since 36, 102, and 84 have a GCF of 6, we can factor out 6:
36n2+102n84=6(6n2)+6(17n)+6(14)=6(6n2+17n14)
To factor a quadratic of the form ax2+bx+c, you need to find two numbers, mm and n, such that mn=ac and m+n=bm+n=b. You can then rewrite the middle term as bx=mx+nx and factor by grouping.
For 6n2+17n14, we then need to find two numbers that multiply to ac=6(14)84 and add to b=17.

Since ac<0ac<0 and b>0b>0, then the two numbers must have opposite signs. Find the pairs of factors of 84 that would give a positive sum and then find the sum of each pair:
Factors Sum 1,841+84=832,422+42

=403,282+28=264,214+21

=176,146+14=87,127+12=5
The pair of −4 and 21 gave a sum of 17 so the middle term can be rewritten as 17n=4n+21n. Factoring by grouping then gives:
6(6n2+17n14) =6(6n24n+21n14)

Rewrite the middle term. =6[(6n24n)+(21n14)]

Group each pair of terms. =6[2n(3n2)+7(3n2)] 

Factor out the GCF of each pair. =6(3n2)(2n+7) 

Factor out 3n2​ ​ ​

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