“Quadratic Equations Part 2 | Roots, Graph & Word Problems Explained”

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📊 Nature of Roots (Very Important)

We use the discriminant:

$D = b^2 – 4ac$

👉 Based on value of D:

• D > 0 → Two different real roots
• D = 0 → Two equal roots
• D < 0 → No real roots (imaginary)

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🔍 Example 1:

$$x^2 – 4x + 4 = 0$$

👉 $D = 16 – 16 = 0$

✔️ Roots are equal
👉 Answer: $x = 2, 2$

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🔍 Example 2:

$$x^2 + 2x + 5 = 0$$

👉 $D = 4 – 20 = -16$

❌ No real roots
✔️ Imaginary roots

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📈 Graph Concept (Parabola)

👉 Every quadratic equation forms a U-shaped graph

$y = ax^2 + bx + c$

• Opens upward if $a > 0$
• Opens downward if $a < 0$

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⭐ Vertex Formula (Very Useful)

$x = \frac{-b}{2a}$

👉 This gives the turning point of the graph

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🎯 Word Problem Example:

The product of two numbers is 12 and their sum is 7. Find the numbers.

👉 Form equation:$$x(7 – x) = 12$$$$x^2 – 7x + 12 = 0$$$$(x – 3)(x – 4) = 0$$

✔️ Numbers are 3 and 4

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🧠 Practice Questions:

1. $x^2 – 6x + 9 = 0$
2. $x^2 + x + 1 = 0$
3. Find vertex of $y = x^2 + 6x + 5$

Learn quadratic equations step-by-step with easy methods, formula, and solved examples for students.