To solve the quadratic equation $-x^2 - 2x = 17$ using the quadratic formula, follow these steps:

Step 1: Rearrange the equation into standard form

The standard form of a quadratic equation is $ax^2 + bx + c = 0$. We need to move all terms to one side of the equation.

$-x^2 - 2x - 17 = 0$

Now the equation is in the form $ax^2 + bx + c = 0$, where:

• $a = -1$
• $b = -2$
• $c = -17$

Step 2: Use the quadratic formula

The quadratic formula is:

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

Substitute the values of $a$, $b$, and $c$:

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

Step 3: Simplify inside the square root

First, calculate the discriminant $\Delta = b^2 - 4ac$:

$\Delta = (-2)^2 - 4(-1)(-17) = 4 - 68 = -64$

Step 4: Compute the roots

Since the discriminant is negative ($\Delta = -64$), the solutions will involve complex numbers. Let's proceed:

$x = \frac{2 \pm \sqrt{-64}}{-2}$

Now, $\sqrt{-64} = 8i$ (where $i$ is the imaginary unit).

$x = \frac{2 \pm 8i}{-2}$

Step 5: Simplify the result

Now divide both terms by $-2$:

$x = -1 \mp 4i$

Thus, the two solutions are:

$x_1 = -1 + 4i$ $x_2 = -1 - 4i$

Final Answer:

The solutions to the equation $-x^2 - 2x = 17$ are:

$x_1 = -1 + 4i, \quad x_2 = -1 - 4i$

Would you like further details on any of the steps?

Here are five related questions to deepen your understanding:

1. How do you interpret complex numbers in real-world applications?
2. What does the discriminant tell you about the nature of quadratic roots?
3. Can you apply the quadratic formula to a real-world scenario involving projectile motion?
4. What is the significance of imaginary numbers in electrical engineering?
5. How would the solution differ if the discriminant were zero?

Tip: When the discriminant is negative, the quadratic equation has two complex conjugate solutions.