Question

solve $x^2 + 12x + 6 = 0$ using the completing-the-square method.
$\bigcirc$ $x = 6 \pm \sqrt{30}$
$\bigcirc$ $x = -6 \pm \sqrt{30}$
$\bigcirc$ $x = 6 \pm \sqrt{6}$
$\bigcirc$ $x = -6 \pm \sqrt{6}$

Explanation:

Step1: Isolate the quadratic and linear terms

Start with the equation \( x^2 + 12x + 6 = 0 \). Subtract 6 from both sides to get \( x^2 + 12x = -6 \).

Step2: Complete the square

The coefficient of \( x \) is 12. Half of 12 is 6, and squaring 6 gives \( 6^2 = 36 \). Add 36 to both sides: \( x^2 + 12x + 36 = -6 + 36 \).

Step3: Rewrite as a perfect square

The left side becomes \( (x + 6)^2 \), and the right side is \( 30 \), so \( (x + 6)^2 = 30 \).

Step4: Solve for x

Take the square root of both sides: \( x + 6 = \pm\sqrt{30} \). Subtract 6 from both sides: \( x = -6 \pm\sqrt{30} \).

Answer:

\( x = -6 \pm\sqrt{30} \) (corresponding to the option: \( x = -6 \pm\sqrt{30} \))