Question

How do I solve a quadratic equation with a negative coefficient?

Answer 1

To solve a quadratic equation, you can follow these steps regardless of whether it has negative coefficients. The general form of a quadratic equation is:

\[
ax^2 + bx + c = 0
\]

where \( a \), \( b \), and \( c \) can be positive or negative.

Here are steps to solve the equation:

### 1. Identify the coefficients
Identify \( a \), \( b \), and \( c \) in your equation.

### 2. Use the Quadratic Formula
The quadratic formula is:

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

### 3. Calculate the Discriminant
Calculate the discriminant \( D = b^2 - 4ac \):
- If \( D > 0 \): You have two distinct real roots.
- If \( D = 0 \): You have one real root (a double root).
- If \( D < 0 \): You have two complex roots.

### 4. Substitute into the Quadratic Formula
Plug the values of \( a \), \( b \), and \( c \) into the quadratic formula and simplify.

### Example
Let's solve the quadratic equation:

\[
-2x^2 + 4x - 6 = 0
\]

1. Here, \( a = -2 \), \( b = 4 \), and \( c = -6 \).

2. Calculate the discriminant:
\[
D = b^2 - 4ac = 4^2 - 4(-2)(-6) = 16 - 48 = -32
\]
Since \( D < 0 \), we know there are two complex solutions.

3. Use the quadratic formula:
\[
x = \frac{{-b \pm \sqrt{D}}}{{2a}} = \frac{{-4 \pm \sqrt{-32}}}{{2 \times -2}}
\]
\[
x = \frac{{-4 \pm \sqrt{32}i}}{{-4}} = \frac{{-4 \pm 4\sqrt{2}i}}{{-4}}
\]
\[
x = 1 \mp \sqrt{2}i
\]

### Final Answer
The solutions to the quadratic equation \(-2x^2 + 4x - 6 = 0\) are:
\[
x = 1 + \sqrt{2}i \quad \text{and} \quad x = 1 - \sqrt{2}i
\]

This method works regardless of the signs of the coefficients.