If x + y = 72\dfrac{7}{2}27 and xy = 52\dfrac{5}{2}25; find:
(i) x - y
(ii) x2 - y2
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(i) By formula,
(x - y)2 = (x + y)2 - 4xy
⇒(x−y)2=(72)2−4×52⇒(x−y)2=494−10⇒(x−y)2=49−404⇒(x−y)2=94⇒(x−y)=94⇒x−y=±32.\Rightarrow (x - y)^2 = \Big(\dfrac{7}{2}\Big)^2 - 4 \times \dfrac{5}{2} \\[1em] \Rightarrow (x - y)^2 = \dfrac{49}{4} - 10 \\[1em] \Rightarrow (x - y)^2 = \dfrac{49 - 40}{4} \\[1em] \Rightarrow (x - y)^2 = \dfrac{9}{4} \\[1em] \Rightarrow (x - y) = \sqrt{\dfrac{9}{4}} \\[1em] \Rightarrow x - y = \pm\dfrac{3}{2}.⇒(x−y)2=(27)2−4×25⇒(x−y)2=449−10⇒(x−y)2=449−40⇒(x−y)2=49⇒(x−y)=49⇒x−y=±23.
Hence, x - y = ±32\pm \dfrac{3}{2}±23.
(ii) Solving,
⇒x2−y2⇒(x−y)(x+y)⇒±32×72⇒±214.\Rightarrow x^2 - y^2 \\[1em] \Rightarrow (x - y)(x + y) \\[1em] \Rightarrow \pm \dfrac{3}{2} \times \dfrac{7}{2} \\[1em] \Rightarrow \pm \dfrac{21}{4}.⇒x2−y2⇒(x−y)(x+y)⇒±23×27⇒±421.
Hence, x2 - y2 = ±214\pm \dfrac{21}{4}±421.
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