# Solve for n (3n+1)^2+6(3n+1)-7=0 (3n+1)2+6(3n+1)-7=0
Factor the left side of the equation.
Let u=3n+1. Substitute u for all occurrences of 3n+1.
u2+6u-7
Factor u2+6u-7 using the AC method.
Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is -7 and whose sum is 6.
-1,7
Write the factored form using these integers.
(u-1)(u+7)
(u-1)(u+7)
Replace all occurrences of u with 3n+1.
(3n+1-1)(3n+1+7)
Simplify.
Combine the opposite terms in 3n+1-1.
Subtract 1 from 1.
(3n+0)(3n+1+7)
3n(3n+1+7)
3n(3n+1+7)
3n(3n+8)
3n(3n+8)
Replace the left side with the factored expression.
3n(3n+8)=0
3n(3n+8)=0
Divide each term in 3n(3n+8)=0 by 3.
3n(3n+8)3=03
Cancel the common factor of 3.
Cancel the common factor.
3n(3n+8)3=03
Divide n(3n+8) by 1.
n(3n+8)=03
n(3n+8)=03
Divide 0 by 3.
n(3n+8)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
n=0
3n+8=0
Set the first factor equal to 0.
n=0
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
3n+8=0
Subtract 8 from both sides of the equation.
3n=-8
Divide each term by 3 and simplify.
Divide each term in 3n=-8 by 3.
3n3=-83
Cancel the common factor of 3.
Cancel the common factor.
3n3=-83
Divide n by 1.
n=-83
n=-83
Move the negative in front of the fraction.
n=-83
n=-83
n=-83
The final solution is all the values that make 3n(3n+8)3=03 true.
n=0,-83
The result can be shown in multiple forms.
Exact Form:
n=0,-83
Decimal Form:
n=0,-2.6‾
Mixed Number Form:
n=0,-223
Solve for n (3n+1)^2+6(3n+1)-7=0

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