4.3 Solution of a Quadratic Equation by Factorisation

Consider the quadratic equation 2x² − 3x + 1 = 0. If we replace x by 1 on the LHS of this equation, we get (2 × 1²) − (3 × 1) + 1 = 0 = RHS of the equation. We say that 1 is a root of the quadratic equation 2x² − 3x + 1 = 0. This also means that 1 is a zero of the quadratic polynomial 2x² − 3x + 1. In general, a real number α is called a root of the quadratic equation ax² + bx + c = 0, a ≠ 0 if aα² + bα + c = 0. We also say that x = α is a solution of the quadratic equation, or that α satisfies the quadratic equation. Note that the zeroes of the quadratic polynomial ax² + bx + c and the roots of the quadratic equation ax² + bx + c = 0 are the same.

You have observed, in Chapter 2, that a quadratic polynomial can have at most two zeroes. So, any quadratic equation can have atmost two roots. You have learnt in Class IX, how to factorise quadratic polynomials by splitting their middle terms. We shall use this knowledge for finding the roots of a quadratic equation. Let us see how. Example: Find the roots of the equation 2x² − 5x + 3 = 0, by factorisation. Solution: Let us first split the middle term − 5x as −2x −3x [because (–2x) × (–3x) = 6x² = (2x²) × 3]. So, 2x² – 5x + 3 = 2x² – 2x – 3x + 3 = 2x (x – 1) –3(x – 1) = (2x – 3)(x – 1). Now, 2x² – 5x + 3 = 0 can be rewritten as (2x – 3)(x – 1) = 0. So, the values of x for which 2x² – 5x + 3 = 0 are the same for which (2x – 3)(x – 1) = 0, i.e., either 2x – 3 = 0 or x – 1 = 0.

Now, 2x – 3 = 0 gives x = 3/2 and x – 1 = 0 gives x = 1. So, x = 3/2 and x = 1 are the solutions of the equation. In other words, 1 and 3/2 are the roots of the equation 2x² – 5x + 3 = 0. Verify that these are the roots of the given equation. Note that we have found the roots of 2x² – 5x + 3 = 0 by factorising 2x² – 5x + 3 into two linear factors and equating each factor to zero. Example: Find the roots of the quadratic equation 6x² – x – 2 = 0. Solution: We have 6x² – x – 2 = 6x² + 3x – 4x – 2 = 3x (2x + 1) – 2 (2x + 1) = (3x – 2)(2x + 1). The roots of 6x² – x – 2 = 0 are the values of x for which (3x – 2)(2x + 1) = 0. Therefore, 3x – 2 = 0 or 2x + 1 = 0, i.e., x = 2/3 or x = –1/2.

Therefore, the roots of 6x² – x – 2 = 0 are 2/3 and –1/2. We verify the roots, by checking that 2/3 and –1/2 satisfy 6x² – x – 2 = 0. Exercise: Find the roots of the following quadratic equations by factorisation: (i) x² – 3x – 10 = 0 (ii) 2x² + x – 6 = 0 (iii) √2x² + 7x + 5√2 = 0 (iv) 2x² – x + 1/8 = 0 (v) 100x² – 20x + 1 = 0. Consider the quadratic equation and the discriminant b² – 4ac which determines whether the equation has real roots or not.