To find the negative solution of the quadratic equation $0 = x^2 + 16x - 17$, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $x^2 + 16x - 17 = 0$:

• $a = 1$
• $b = 16$
• $c = -17$

Now, plug these values into the quadratic formula:

$x = \frac{-16 \pm \sqrt{16^2 - 4(1)(-17)}}{2(1)}$

$x = \frac{-16 \pm \sqrt{256 + 68}}{2}$

$x = \frac{-16 \pm \sqrt{324}}{2}$

$x = \frac{-16 \pm 18}{2}$

Now, compute the two possible values of $x$:

1. $x = \frac{-16 + 18}{2} = \frac{2}{2} = 1$
2. $x = \frac{-16 - 18}{2} = \frac{-34}{2} = -17$

The negative solution is $x = -17$.

Would you like more details or have any questions? Here are some related questions:

1. How do you derive the quadratic formula?
2. What are the steps for completing the square in a quadratic equation?
3. How can you determine the nature of the roots of a quadratic equation?
4. What is the discriminant, and how is it used in quadratic equations?
5. How can you solve quadratic equations by factoring?

Tip: Always check the discriminant ($b^2 - 4ac$) before solving a quadratic equation to determine if the roots are real or complex.