What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree 2:
$$
ax^2 + bx + c = 0, \quad a \neq 0
$$
Quadratic Formula
The roots of a quadratic equation are given by:
$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$
Discriminant
The discriminant $D = b^2 - 4ac$ determines the nature of roots:
| Discriminant | Nature of Roots |
|---|---|
| $D > 0$ | Two distinct real roots |
| $D = 0$ | Two equal real roots |
| $D < 0$ | No real roots (complex) |
Methods of Solving
1. Factorization Method
Example: Solve $x^2 - 5x + 6 = 0$
$$
x^2 - 5x + 6 = 0
$$
$$
(x - 2)(x - 3) = 0
$$
$$
x = 2 \text{ or } x = 3
$$
2. Completing the Square
Example: Solve $x^2 + 6x + 2 = 0$
$$
x^2 + 6x = -2
$$
$$
x^2 + 6x + 9 = -2 + 9
$$
$$
(x + 3)^2 = 7
$$
$$
x = -3 \pm \sqrt{7}
$$
3. Using the Quadratic Formula
Example: Solve $2x^2 + 3x - 5 = 0$
Here, $a = 2$, $b = 3$, $c = -5$
$$
D = 9 - 4(2)(-5) = 9 + 40 = 49
$$
$$
x = \frac{-3 \pm 7}{4}
$$
$$
x = 1 \text{ or } x = -\frac{5}{2}
$$
Sum and Product of Roots
If $\alpha$ and $\beta$ are roots of $ax^2 + bx + c = 0$:
$$
\alpha + \beta = -\frac{b}{a}
$$
$$
\alpha \cdot \beta = \frac{c}{a}
$$
Practice Problems
- Find the roots of $x^2 - 7x + 12 = 0$
- Determine the nature of roots of $3x^2 + 2x + 1 = 0$
- If $\alpha$ and $\beta$ are roots of $x^2 - 5x + 6 = 0$, find $\alpha^2 + \beta^2$