Questions: Solve the quadratic equation by completing the square and applying the square root property. Write imaginary solutions in the form (a pm b i). (m^2+4 m+6=0) The solution set is (square) (). (square)

Solve the quadratic equation by completing the square and applying the square root property. Write imaginary solutions in the form (a pm b i).

(m^2+4 m+6=0)

The solution set is (square) (). (square)

Transcript text: Solve the quadratic equation by completing the square and applying the square root property. Write imaginary solutions in the form $a \pm b i$. \[ m^{2}+4 m+6=0 \] The solution set is $\square$ $\}$. $\square$

Solution

Solution Steps

Step 1: Completing the Square

To solve the quadratic equation $m^2 + 4m + 6 = 0$, we first complete the square. We rewrite the equation as: \[ (m + 2)^2 = 2 - 6 \] This simplifies to: \[ (m + 2)^2 = -2 \]

Step 2: Applying the Square Root Property

Next, we apply the square root property. Taking the square root of both sides gives us: \[ m + 2 = \pm \sqrt{-2} \] This can be expressed in terms of imaginary numbers: \[ m + 2 = \pm i\sqrt{2} \]

Step 3: Solving for $m$

Now, we isolate $m$: \[ m = -2 \pm i\sqrt{2} \]