### Solving Quadratic Equations for JEE Mains

For JEE Mains, mastering quadratic equations is crucial as they form the basis of many problems. Here's a detailed guide on solving quadratic equations tailored for JEE Mains:

### Key Concepts

1. **Standard Form:**
   \[ ax^2 + bx + c = 0 \]

2. **Roots of the Quadratic Equation:**
   If \(\alpha\) and \(\beta\) are the roots, then:
   \[ \alpha + \beta = -\frac{b}{a} \]
   \[ \alpha \beta = \frac{c}{a} \]

3. **Discriminant:**
   \[ \Delta = b^2 - 4ac \]
   - \(\Delta > 0\): Two distinct real roots.
   - \(\Delta = 0\): One real root (repeated).
   - \(\Delta < 0\): No real roots (complex roots).

### Methods to Solve

#### 1. Factoring

Factor the quadratic expression if possible.

Example:
\[ 2x^2 - 8x + 6 = 0 \]

1. Find factors of the form \( (2x + p)(x + q) \).
2. After solving, we get:
   \[ (2x - 2)(x - 3) = 0 \]
3. Roots: \( x = 1 \) and \( x = 3 \).

#### 2. Quadratic Formula

\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]

Example:
\[ x^2 - 4x - 5 = 0 \]

1. Calculate the discriminant:
   \[ \Delta = (-4)^2 - 4 \cdot 1 \cdot (-5) = 16 + 20 = 36 \]
2. Use the formula:
   \[ x = \frac{4 \pm \sqrt{36}}{2} = \frac{4 \pm 6}{2} \]
3. Roots: \( x = 5 \) and \( x = -1 \).

#### 3. Completing the Square

Rewriting the equation to form a perfect square trinomial.

Example:
\[ x^2 + 6x + 8 = 0 \]

1. Rewrite:
   \[ x^2 + 6x = -8 \]
2. Add \((\frac{6}{2})^2\):
   \[ x^2 + 6x + 9 = 1 \]
3. Form the square:
   \[ (x + 3)^2 = 1 \]
4. Solve:
   \[ x + 3 = \pm 1 \]
   \[ x = -2 \text{ and } x = -4 \]

### Example Problems for Practice

1. Solve \(3x^2 + 5x - 2 = 0\) using the quadratic formula.
2. Factor and solve \(x^2 - 7x + 12 = 0\).
3. Complete the square to solve \(x^2 + 8x + 15 = 0\).
4. Find the nature of the roots of \(2x^2 + 3x + 1 = 0\).
5. Solve \(x^2 - 2x - 8 = 0\) by factoring.
6. Use the quadratic formula to solve \(4x^2 + 4x + 1 = 0\).
7. Determine the roots of \(5x^2 - 6x + 1 = 0\).
8. Graph and find the roots of \(x^2 - x - 2 = 0\).

### Tip

For JEE Mains, practice solving quadratic equations quickly and accurately. Ensure you're comfortable with all methods and can identify the best approach for each problem. 

Do you want more details on any specific method or have any questions?

Master solving quadratic equations for JEE Mains with a detailed guide on key concepts like the quadratic formula, factoring, and completing the square. Practice example problems to enhance your preparation.

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