Question

solve the quadratic equation by completing the square and applying the square root property. write imaginary solutions in the form $a \pm bi$.
$n^2 + 6n + 8 = 0$
the solution set is \\{\\}.

Explanation:

Step1: Isolate the variable terms

Start with the equation \( n^2 + 6n + 8 = 0 \). Subtract 8 from both sides to get \( n^2 + 6n = -8 \).

Step2: Complete the square

Take half of the coefficient of \( n \), which is \( \frac{6}{2}=3 \), and square it: \( 3^2 = 9 \). Add 9 to both sides of the equation: \( n^2 + 6n + 9 = -8 + 9 \). The left side is now a perfect square trinomial, so it factors to \( (n + 3)^2 = 1 \).

Step3: Apply the square root property

Take the square root of both sides: \( n + 3 = \pm\sqrt{1} \), which simplifies to \( n + 3 = \pm 1 \).

Step4: Solve for \( n \)

For the positive case: \( n + 3 = 1 \), subtract 3 from both sides: \( n = 1 - 3 = -2 \). For the negative case: \( n + 3 = -1 \), subtract 3 from both sides: \( n = -1 - 3 = -4 \).

Answer:

\(\{-4, -2\}\)