The quadratic formula is a solution for finding the roots of a quadratic equation of the form:

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

where:
- \( a \), \( b \), and \( c \) are constants, and
- \( a \neq 0 \) (since the equation would not be quadratic if \( a = 0 \)).

The formula is:

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

### Explanation:
- **Discriminant (\( b^2 - 4ac \))**: This part under the square root 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 \( b^2 - 4ac < 0 \), there are two complex roots.

The \( \pm \) symbol means there are two solutions for \( x \), calculated by adding and subtracting the square root of the discriminant.

Would you like more details or have any specific questions?

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### Related Questions:
1. How is the discriminant used to determine the nature of the roots?
2. What are some real-life applications of the quadratic formula?
3. How can we derive the quadratic formula by completing the square?
4. Why does the formula not work if \( a = 0 \)?
5. How is the quadratic formula used in solving equations in physics?

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**Tip:** Memorize the quadratic formula by practicing with different values of \( a \), \( b \), and \( c \).

Explore the quadratic formula and learn how to find the roots of a quadratic equation with step-by-step explanations. This guide covers essential algebraic concepts, including the discriminant and the nature of roots, tailored for students in Grades 8-10.

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