The quadratic formula is a method used to solve quadratic equations of the form:

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

Here, \( a \), \( b \), and \( c \) are coefficients, with \( a \neq 0 \). The quadratic formula provides the solutions for \( x \) as follows:

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

### Components of the Formula:
1. **\( b^2 - 4ac \)**: This is called the **discriminant**. It determines the nature of the roots:
   - If \( b^2 - 4ac > 0 \), there are two distinct real roots.
   - If \( b^2 - 4ac = 0 \), there is exactly one real root (a repeated root).
   - If \( b^2 - 4ac < 0 \), the roots are complex and not real.

2. **\( \pm \)**: This indicates that there are generally two solutions for \( x \), one involving addition and the other involving subtraction.

### Example:
Let's solve the quadratic equation \( 2x^2 - 4x - 6 = 0 \) using the quadratic formula.

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

2. Substitute these values into the quadratic formula:
   \[
   x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-6)}}{2(2)}
   \]
   \[
   x = \frac{4 \pm \sqrt{16 + 48}}{4}
   \]
   \[
   x = \frac{4 \pm \sqrt{64}}{4}
   \]
   \[
   x = \frac{4 \pm 8}{4}
   \]

3. Solve for the two possible values of \( x \):
   - \( x = \frac{4 + 8}{4} = \frac{12}{4} = 3 \)
   - \( x = \frac{4 - 8}{4} = \frac{-4}{4} = -1 \)

So, the solutions are \( x = 3 \) and \( x = -1 \).

Would you like to dive deeper into any particular aspect of the quadratic formula, or do you have any related questions?

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Here are 5 additional questions related to the quadratic formula:

1. How would you use the quadratic formula to solve \( 3x^2 + 5x - 2 = 0 \)?
2. What does the discriminant tell you about the nature of the roots before solving a quadratic equation?
3. Can you derive the quadratic formula by completing the square on the general quadratic equation?
4. How would you solve a quadratic equation where the discriminant is negative, resulting in complex roots?
5. What are the implications of a zero discriminant in a real-world context, such as physics or engineering?

**Tip:** Always simplify the quadratic equation as much as possible before applying the quadratic formula, and check your discriminant to anticipate the type of roots you'll find.

Learn how to solve the quadratic equation 3x^2 + 5x - 2 using the quadratic formula with a detailed step-by-step solution. This problem covers key algebraic concepts, including the discriminant and completing the square. Suitable for high school students.

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