To find the roots of the quadratic equation $-x^2 - 4x + 5 = 0$, you can either factor the equation or use the quadratic formula. Since this equation does not factor easily, let's use the quadratic formula:

The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$. For the given equation $-x^2 - 4x + 5 = 0$, we have:

• $a = -1$ (coefficient of $x^2$),
• $b = -4$ (coefficient of $x$),
• $c = 5$ (constant term).

Step-by-step solution:

1. Substitute $a = -1$, $b = -4$, and $c = 5$ into the quadratic formula:

   $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(-1)(5)}}{2(-1)}$

2. Simplify the expression:

   $x = \frac{4 \pm \sqrt{16 + 20}}{-2}$

   $x = \frac{4 \pm \sqrt{36}}{-2}$

3. Solve the square root of 36:

   $x = \frac{4 \pm 6}{-2}$

4. Find the two possible solutions:

   • First solution: $x = \frac{4 + 6}{-2} = \frac{10}{-2} = -5$

   • Second solution: $x = \frac{4 - 6}{-2} = \frac{-2}{-2} = 1$

The roots are:

$x = -5 \quad \text{or} \quad x = 1$

Let me know if you'd like more details or have any questions!

Here are 5 related questions to expand your understanding:

1. How would the quadratic formula change if $b$ were positive?
2. How do you determine the nature of the roots using the discriminant?
3. Can you solve this quadratic equation by completing the square?
4. What does the graph of $-x^2 - 4x + 5$ look like?
5. How would you solve $-x^2 - 4x + 5 > 0$ (an inequality)?

Tip: Always check the discriminant $b^2 - 4ac$ before solving a quadratic equation. It tells you how many and what kind of roots you will get!