Question

How to solve a quadratic equation of the form ax^2 + bx + c = 0

Answer 1

To solve a quadratic equation of the form \( ax^2 + bx + c = 0 \), you can use various methods such as factoring, completing the square, or the quadratic formula. The quadratic formula is the most general method and can be applied to any quadratic equation. Here’s how to use it:

### Quadratic Formula
The quadratic formula is given by:

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

### Steps to Solve Using the Quadratic Formula

1. **Identify the coefficients**: Identify the values of \( a \), \( b \), and \( c \) from your equation \( ax^2 + bx + c = 0 \).

2. **Calculate the discriminant**: The discriminant is given by \( D = b^2 - 4ac \). The value of the discriminant will tell you about the nature of the roots:
- If \( D > 0 \), there are two distinct real roots.
- If \( D = 0 \), there is exactly one real root (a repeated root).
- If \( D < 0 \), there are two complex (non-real) roots.

3. **Apply the quadratic formula**: Plug the values of \( a \), \( b \), and \( D \) into the quadratic formula.

4. **Simplify**: Compute the values of \( x \). You’ll typically end up with two values based on the \( \pm \) in the formula.

### Example
Solve the equation \( 2x^2 + 3x - 2 = 0 \).

1. **Identify coefficients**: Here, \( a = 2 \), \( b = 3 \), \( c = -2 \).

2. **Calculate the discriminant**:
\[
D = b^2 - 4ac = 3^2 - 4(2)(-2) = 9 + 16 = 25
\]
Since \( D > 0 \), there will be two distinct real roots.

3. **Apply the quadratic formula**:
\[
x = \frac{-3 \pm \sqrt{25}}{2 \cdot 2} = \frac{-3 \pm 5}{4}
\]

4. **Calculate the roots**:
- For \( x_1 = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2} \)
- For \( x_2 = \frac{-3 - 5}{4} = \frac{-8}{4} = -2 \)

### Final Solutions
The solutions of the equation \( 2x^2 + 3x - 2 = 0 \) are \( x = \frac{1}{2} \) and \( x = -2 \).