Factorise :
14(a+b)2−916(2a−b)2\dfrac{1}{4}(a + b)^2 - \dfrac{9}{16}(2a - b)^241(a+b)2−169(2a−b)2
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Given,
=14(a+b)2−916(2a−b)2=[12(a+b)]2−[34(2a−b)]2=[12(a+b)+34(2a−b)][12(a+b)−34(2a−b)]=[2(a+b)+3(2a−b)4][2(a+b)−3(2a−b)4]=(2a+2b+6a−3b4)(2a+2b−6a+3b4)=8a−b4×5b−4a4=116(8a−b)(5b−4a).\phantom{=}\dfrac{1}{4}(a + b)^2 - \dfrac{9}{16}(2a - b)^2 \\[1em] = \Big[\dfrac{1}{2}(a + b)\Big]^2 - \Big[\dfrac{3}{4}(2a - b)\Big]^2 \\[1em] = \Big[\dfrac{1}{2}(a + b) + \dfrac{3}{4}(2a - b)\Big]\Big[\dfrac{1}{2}(a + b) - \dfrac{3}{4}(2a - b)\Big] \\[1em] = \Big[\dfrac{2(a + b) + 3(2a - b)}{4}\Big]\Big[\dfrac{2(a + b) - 3(2a - b)}{4}\Big] \\[1em] = \Big(\dfrac{2a + 2b + 6a - 3b}{4}\Big)\Big(\dfrac{2a + 2b - 6a + 3b}{4}\Big)\\[1em] = \dfrac{8a - b}{4} \times \dfrac{5b - 4a}{4} \\[1em] = \dfrac{1}{16}(8a - b)(5b - 4a).=41(a+b)2−169(2a−b)2=[21(a+b)]2−[43(2a−b)]2=[21(a+b)+43(2a−b)][21(a+b)−43(2a−b)]=[42(a+b)+3(2a−b)][42(a+b)−3(2a−b)]=(42a+2b+6a−3b)(42a+2b−6a+3b)=48a−b×45b−4a=161(8a−b)(5b−4a).
Hence, 14(a+b)2−916(2a−b)2=116(8a−b)(5b−4a).\dfrac{1}{4}(a + b)^2 - \dfrac{9}{16}(2a - b)^2 = \dfrac{1}{16}(8a - b)(5b - 4a).41(a+b)2−169(2a−b)2=161(8a−b)(5b−4a).
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Evaluate :
5.67×5.67×5.67+4.33×4.33×4.335.67×5.67−5.67×4.33+4.33×4.33\dfrac{5.67 \times 5.67 \times 5.67 + 4.33 \times 4.33 \times 4.33}{5.67 \times 5.67 - 5.67 \times 4.33 + 4.33 \times 4.33}5.67×5.67−5.67×4.33+4.33×4.335.67×5.67×5.67+4.33×4.33×4.33
9x2 + 3x - 8y - 64y2
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