Ex 4.3, 1

Transcript

Ex 4.3 ,1 Find the roots of the following quadratic equations, if they exist, by the method of completing the square: (i) 2x2 – 7x + 3 = 0 2x2 – 7x +3 = 0 Dividing by 2 (2𝑥2 − 7𝑥 + 3 = 0)/2=0/2 2𝑥2/2 – 7𝑥/2 + 3/2=0 x2 – 7𝑥/2+3/2=0 We know that (a – b)2 = a2 – 2ab + b2 Here, a = x & – 2ab = – 7𝑥/2 – 2xb = −7𝑥/2 b = −7𝑥/(2(−2𝑥)) b = 7/4 Now, in our equation x2 −7𝑥/2+3/2=0 Adding and subtracting (7/4)^2 x2 −7𝑥/2+3/2+(7/4)^2−(7/4)^2=0 x2 +(7/4)^2−7𝑥/2+3/2−(7/4)^2=0 (𝑥− 7/4)^2+3/2 −(7/4)^2=0 (𝑥− 7/4)^2+3/2−49/16=0 (𝑥− 7/4)^2+(3(8) − 49)/16=0 (𝑥− 7/4)^2+(24 − 49)/16=0  (𝑥− 7/4)^2−25/16=0 (𝑥−7/4)^2=25/16 (𝑥−7/4)^2=(5/4)^2 Cancelling square both sides 𝑥−7/4 = ± 5/4 Ex 4.3 ,1 Find the roots of the following quadratic equations, if they exist, by the method of completing the square: (ii) 2x2 + x – 4 = 0 2x2 + x – 4 = 0 Dividing whole equation by 2 (2𝑥2 + 𝑥 − 4)/2=0/2 2𝑥2/2+𝑥/2−4/2=0 x2 + 𝑥/2−2=0 We know that (a + b)2 = a2 + 2ab + b2 Here, a = x & 2ab = 𝑥/2 2xb = 𝑥/2  2b = 1/2 b = 1/2 × 1/2 b = 1/4 Now, in our equation x2 + 𝑥/2−2=0 Adding and subtracting (1/4)^2 x2 + 𝑥/2−2+(1/4)^2−(1/4)^2= 0 x2 + 𝑥/2+(1/4)^2– 2 – (1/4)^2=0 (𝑥+1/4)^2−2−(1/4)^2= 0 (𝑥+1/4)^2−2 −1/16=0 (𝑥+1/4)^2=2+1/16  (𝑥+1/4)^2=(2(16) + 1)/16 (𝑥+1/4)^2=(32 + 1)/16 (𝑥+1/4)^2=33/16 (𝑥+1/4)^2=(√33/4)^2 Cancelling square both sides 𝑥+1/4 = ± √33/4 Solving  So, the root of the equation are x = (√33 − 1)/4 & x = (−(√33 + 1))/4 Ex 4.3 ,1 Find the roots of the following quadratic equations, if they exist, by the method of completing the square: (iii) 4x2 + 4√3 𝑥+3=0 4x2 + 4 √3 𝑥+3=0 Dividing whole equation 4 (4𝑥^2+ 4 √3 𝑥+ 3)/4=0/4 (4𝑥^2)/4 + (4 √3)/4 x + 3/4=0 x2 + √3 𝑥+ 3/4 = 0 We know that (a + b)2 = a2 + 2ab + b2 Here, a = x & 2ab = √3 𝑥 2xb = √3 𝑥  2b = √3 b = √3/2 Now, in our equation x2 + √3 𝑥+3/4=0 Adding and subtracting (√3/2)^2 x2 + √3 𝑥+3/4+(√3/2)^2−(√3/2)^2=0 x2 + √3 𝑥+(√3/2)^2+3/4−(√3/2)^2=0 (𝑥+√3/2 )^2+3/4−(√3/2)^2=0 (𝑥+√3/2 )^2+3/4−3/4=0 (𝑥+√3/2 )^2=0 (𝑥+√3/2 )^2=02 Cancelling square both sides (𝑥+√3/2 )^2= ± 0 So, the root of the equation are x = (−√3)/2 & x = (−√3)/2 Ex 4.3 ,1 Find the roots of the following quadratic equations, if they exist, by the method of completing the square: (iv) 2x2 + x + 4 = 0 2x2 + x + 4 = 0 Dividing equation by 2 (2𝑥2 + 𝑥 + 4)/2=0/2 2𝑥2/2+𝑥/2 + 4/2=0 x2 + 𝑥/2+2=0 We know that (a + b)2 = a2 + 2ab + b2 Here, a = x & 2ab = 𝑥/2 2xb = 𝑥/2  2b = 1/2 b = 1/2×1/2 b = 1/4 Now, in our equation x2 + 𝑥/2+2=0 Adding and subtracting (1/4)^2 x2 + 𝑥/2 +2+(1/4)^2−(1/4)^2=0 "x2 + " 𝑥/2+(1/4)^2+2−(1/4)^2=0 (𝑥+1/4)^2+2−(1/4)^2=0 (𝑥+1/4)^2+2−1/16=0 (𝑥+1/4)^2+((32 − 1)/16)=0  (𝑥+1/4)^2+ 31/16=0 (𝑥+1/4)^2=(−31)/16 Since square of any number cannot be negative So, answer does not exist