Example 2 - Chapter 4 Class 10 Quadratic Equations (Important Question)

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Example 2 Check whether the following are quadratic equations: (i) (x – 2)2 + 1 = 2x – 3 (x – 2)2 + 1 = 2x – 3 Using (a – b)2 = a2 + b2 – 2ab (x2 + 4 – 4x) + 1 = 2x – 3 x2 + 5 – 4x = 2x – 3 x2 + 5 – 4x – 2x + 3 = 0 x2 – 6x + 8 = 0 It is the form ax2 + b x + c = 0 Where, a = 1, b = – 6, c = 8 Hence, it is a quadratic equation . Example 2 Check whether the following are quadratic equations: (ii) x(x + 1) + 8 = (x + 2) (x – 2) x (x + 1) + 8 = (x + 2) (x – 2) Using (a + b) (a – b) = a2 – b2 x (x + 1) + 8 = x2 – 4 x2 + x + 8 = x2 – 4 x2 + x + 8 – x2 + 4 = 0 (x2 – x2) + x + 8 + 4 = 0 x + 12 = 0 Since the highest power is 1 not 2 It is not in the form of 𝑎𝑥2 + 𝑏𝑥 + 𝑐 =0 Hence, it is not a quadratic equation . Example 2 Check whether the following are quadratic equations: (iii) x (2x + 3) = x2 + 1 x(2x + 3) = x2 + 1 2x2 + 3x = x2 + 1 2x2 + 3x – x2 – 1 = 0 (2x2 – x2) + 3x – 1 = 0 x2 + 3x – 1 = 0 It is the form of ax2 + bx + c = 0 Where a = 1, b = 3, c = – 1 Hence, it is a quadratic equation . Example 2 Check whether the following are quadratic equations: (iv) (x + 2)3 = x3 – 4 (x + 2)3 = x3 – 4 Using (a + b)3 = a3 + b3 + 3a2b + 3ab2 x3 + 23 + 3 (x2) (2) + 3 (x) (2)^2 = x3 – 4 x3 + 8 + 6x2 + 12x = x3 – 4 x3 + 8 + 6x2 + 12 x – x3 + 4 = 0 6x2 + 12x + 12 = 0 6(x2 + 2x + 2) = 0 x2 + 2x + 2 = 0 It is of the form ax2 + bx + c = 0 Where a = 1, b = 2, c = 2 Hence, it is quadratic equation .