Quadratic Equations | Class 10 Mathematics Notes

4.3 Solution of a Quadratic Equation by Factorisation

This section explains how to find the roots (solutions) of a quadratic equation by factorisation. A root α of the quadratic equation ax² + bx + c = 0 is a real number such that aα² + bα + c = 0. The roots of the quadratic equation correspond to the zeroes of the quadratic polynomial ax² + bx + c.

Since a quadratic polynomial can have at most two zeroes, a quadratic equation can have at most two roots. The method of factorisation involves expressing the quadratic polynomial as a product of two linear factors and then setting each factor equal to zero to find the roots.

The section provides detailed examples illustrating this method:

• Example 3: Solve 2x² - 5x + 3 = 0 by splitting the middle term -5x into -2x and -3x, factorising as (2x - 3)(x - 1) = 0, giving roots x = 3/2 and x = 1.

• Example 4: Solve 6x² - x - 2 = 0 by splitting -x into 3x and -4x, factorising as (3x - 2)(2x + 1) = 0, giving roots x = 2/3 and x = -1/2.

• Example 5: Solve 3x² - 2√6 x + 2 = 0 by expressing as (√3 x - √2)² = 0, giving a repeated root x = √(2/3).

• Example 6: Apply factorisation to the prayer hall problem's quadratic equation 2x² + x - 300 = 0, factorising as (x - 12)(2x + 25) = 0, discarding the negative root, and concluding the breadth is 12 m and length 25 m.

The section emphasizes verifying roots by substituting back into the original equation and highlights that factorisation is a powerful method when applicable.

📊 Diagram: Fig. 4.1

🧪 Activity: Example 3-6 illustrate factorisation method for solving quadratic equations.

🔗 Connection: After learning factorisation, the next section discusses the nature of roots using the discriminant.