Rewrite each equation below. Then solve your new equation. Be sure to check your solution using the original equation.
(n+4)+n(n+2)+n=0
(n+4)+n(n+2)+n=0
(n+4)+n(n+2+1)=0
(n+4)+n(n+3) = 0
n+4+n^2+3n = 0
n^2+4n+4 = 0
(n+2)^2 = 0
n = -2
or,
n+2+2 + n(n+2) + n = 0
(n+2)+n(n+2) + n+2 = 0
2(n+2)+n(n+2) = 0
(n+2)(n+2) = 0
(n+2)^2 = 0
To rewrite the given equation, let's simplify it step-by-step:
Step 1: Distribute the term 'n' to the terms inside the parentheses:
n + 4 + n^2 + 2n + n = 0
Step 2: Combine like terms on the left side:
n^2 + 4n + 4 = 0
Now, we have the new equation: n^2 + 4n + 4 = 0.
To solve this quadratic equation, we can apply the quadratic formula:
The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 1, b = 4, and c = 4.
Now, let's substitute these values into the quadratic formula and solve for 'n':
n = (-4 ± √(4^2 - 4(1)(4))) / (2 * 1)
n = (-4 ± √(16 - 16)) / 2
n = (-4 ± √0) / 2
n = (-4 ± 0) / 2
Since the square root of 0 is 0, the expression simplifies to:
n = -4 / 2
n = -2
Therefore, the solution to the new equation is n = -2.
Now, let's check this solution using the original equation:
Original equation: (n+4) + n(n+2) + n = 0
Substituting n = -2:
(-2 + 4) + (-2)(-2 + 2) + (-2) = 0
(2) + (-2)(0) + (-2) = 0
2 + 0 + (-2) = 0
0 = 0
The left side of the original equation equals the right side, so n = -2 is indeed the correct solution.
Therefore, the solution to the original equation is n = -2.