Question

solve for k by completing the square.
$k^2 + k - 6 = 0$
show your work here

Explanation:

Step1: Move constant to right

Start with the equation \( k^2 + k - 6 = 0 \). Add 6 to both sides: \( k^2 + k = 6 \).

Step2: Complete the square

For \( k^2 + k \), take half of the coefficient of \( k \) (which is \( \frac{1}{2} \)), square it (\( (\frac{1}{2})^2 = \frac{1}{4} \)), and add to both sides: \( k^2 + k + \frac{1}{4} = 6 + \frac{1}{4} \).
The left side becomes a perfect square: \( (k + \frac{1}{2})^2 = \frac{25}{4} \).

Step3: Solve for k

Take square roots: \( k + \frac{1}{2} = \pm \frac{5}{2} \).
Subtract \( \frac{1}{2} \): \( k = -\frac{1}{2} \pm \frac{5}{2} \).
Calculate two solutions:

• \( k = -\frac{1}{2} + \frac{5}{2} = 2 \)
• \( k = -\frac{1}{2} - \frac{5}{2} = -3 \)

Answer:

\( k = 2 \) and \( k = -3 \)