Solve -x^2-2x=17 using the Quadratic Formula.

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.

Solve -x^2 - 2x = 17 using the Quadratic Formula.

This detailed guide walks through solving the quadratic equation -x^2 - 2x = 17 using the quadratic formula. It explains each step, including rearranging the equation, using the discriminant, and working with complex numbers. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation