Quadratic Equations | Class 10 Mathematics Notes

4.4 Nature of Roots

This section introduces the discriminant of a quadratic equation and explains how it determines the nature of the roots. The quadratic formula for roots of ax² + bx + c = 0 is given by x = (-b ± √(b² - 4ac)) / (2a). The expression under the square root, Δ = b² - 4ac, is called the discriminant.

The nature of roots depends on the value of the discriminant:

1. If Δ > 0, the equation has two distinct real roots given by (-b + √Δ)/(2a) and (-b - √Δ)/(2a).

2. If Δ = 0, the equation has two equal real roots (repeated root) both equal to -b/(2a).

3. If Δ < 0, there are no real roots because the square root of a negative number is not real.

The section provides examples to illustrate these cases:

• Example 7: For 2x² - 4x + 3 = 0, Δ = 16 - 24 = -8 < 0, so no real roots.

• Example 8: A pole is to be erected on the boundary of a circular park of diameter 13 m such that the difference of distances from two opposite gates is 7 m. Using Pythagoras theorem and setting up the quadratic equation x² + 7x - 60 = 0, the discriminant is 289 > 0, so two real roots exist. The positive root x = 5 m is chosen, giving the distances from gates as 5 m and 12 m.

• Example 9: For 3x² - 2x + 1/3 = 0, Δ = 4 - 4 = 0, so two equal real roots x = 1/3.

This section emphasizes the importance of the discriminant in predicting the number and type of roots before solving the equation.

📊 Diagram: Fig. 4.2

🧪 Activity: Example 8 involves applying the discriminant to a real-life problem involving distances and a circle.

🔗 Connection: This section prepares students to use the quadratic formula and understand root nature, leading to exercises and summary.