To solve the quadratic equation $z^2 + 26z + 49 = 0$ by completing the square, follow these steps:

Step 1: Move the constant term to the other side.

$z^2 + 26z = -49$

Step 2: Complete the square.

To complete the square, take half of the coefficient of $z$ (which is 26), square it, and add it to both sides.

$\left(\frac{26}{2}\right)^2 = 13^2 = 169$

Add 169 to both sides of the equation:

$z^2 + 26z + 169 = -49 + 169$

$z^2 + 26z + 169 = 120$

Step 3: Rewrite the left side as a perfect square.

The left-hand side is a perfect square trinomial:

$(z + 13)^2 = 120$

Step 4: Solve for $z$.

Take the square root of both sides:

$z + 13 = \pm \sqrt{120}$

$z + 13 = \pm 10.95 \quad \text{(rounded to two decimal places)}$

Step 5: Isolate $z$.

Now solve for $z$ by subtracting 13 from both sides:

$z = -13 \pm 10.95$

So, we have two solutions:

$z_1 = -13 + 10.95 = -2.05$

$z_2 = -13 - 10.95 = -23.95$

Final Answer:

$z = -2.05 \quad \text{or} \quad z = -23.95$

Would you like further details or clarification on any of the steps?

5 Related Questions:

1. How do you solve quadratic equations using the quadratic formula?
2. What are the other methods to solve quadratic equations besides completing the square?
3. How is the discriminant related to the number of solutions of a quadratic equation?
4. How do you graph a quadratic function and interpret its roots?
5. What is the difference between rational and irrational roots in quadratic equations?

Tip:

Completing the square is especially useful for solving quadratic equations and for deriving the quadratic formula!