Solve the equation by completing the square.

Question

zsquaredplus 10zplus 6equals0
Question content area bottom
Part 1
A.5 plus or minus nothing StartRoot 6 EndRoot
5plus or minusnothingStartRoot 6 EndRoot
B.5 plus StartRoot 19 EndRoot
5plusStartRoot 19 EndRoot
C.minus 5 plus or minus StartRoot 19 EndRoot
minus5plus or minusStartRoot 19 EndRoot
D.minus 10 plus nothing StartRoot 6 EndRoot
minus10plusnothingStartRoot 6 EndRoot

Answer 1

To solve the equation \( z^2 + 10z + 6 = 0 \) by completing the square, follow these steps:

1. Rearrange the equation into a suitable form:

   \[ z^2 + 10z = -6 \]

2. Complete the square for the left side. To do this, take half of the coefficient of \( z \) (which is 10), square it, and add it to both sides.

   • Half of 10 is 5.
   • Squaring 5 gives \( 25 \).

   So, we add \( 25 \) to both sides:

   \[ z^2 + 10z + 25 = -6 + 25 \] \[ z^2 + 10z + 25 = 19 \]

3. Factor the left side:

   \[ (z + 5)^2 = 19 \]

4. Take the square root of both sides:

   \[ z + 5 = \pm \sqrt{19} \]

5. Solve for \( z \):

   \[ z = -5 \pm \sqrt{19} \]

Thus, the solutions to the equation \( z^2 + 10z + 6 = 0 \) are:

\[ z = -5 + \sqrt{19} \quad \text{and} \quad z = -5 - \sqrt{19} \]

Among the provided options, the correct answer is:

C. \( -5 \pm \sqrt{19} \)