Question

ifx+y=7/2 and xy=5/2: find x2-y2
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Answer 1

$\huge\purple{\underline{\underline{\pink{Ans}\red{wer:-}}}}$

$\sf{The \ value \ of \ x^{2}-y^{2} \ is \ \frac{29}{4}.}$

$\sf\orange{Given:}$

$\sf{\implies{x+y=\frac{7}{2}}}$

$\sf{\implies{xy=\frac{5}{2}}}$

$\sf\pink{To \ find:}$

$\sf{The \ value \ of \ x^{2}-y^{2}}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{\implies{x+y=\frac{7}{2}...(1)}}$

$\sf{\implies{xy=\frac{5}{2}...(2)}}$

$\sf{According \ to \ identity}$

$\sf{(a-b)^{2}=(a+b)^{2}-4ab}$

$\sf{\implies{(x-y)^{2}=(x+y)^{2}-4xy}}$

$\sf{\implies{(x-y)^{2}=(\frac{7}{2})^{2}-4×\frac{5}{2}}}$

$\sf{\implies{(x-y)^{2}=\frac{49}{4}-\frac{20}{2}}}$

$\sf{\implies{(x-y)^{2}=\frac{49-40}{4}}}$

$\sf{\implies{(x-y)^{2}=\frac{9}{4}}}$

$\sf{Taking \ square \ root \ of \ both \ sides}$

$\sf{\implies{x-y=\frac{3}{2}...(3)}}$

$\sf{Add \ equations \ (1) \ and \ (3)}$

$\sf{x+y=\frac{7}{2}}$

$\sf{+}$

$\sf{x-y=\frac{3}{2}}$

______________________

$\sf{2x=\frac{7}{2}+\frac{3}{2}}$

$\sf{2x=\frac{10}{2}}$

$\sf{2x=5}$

$\sf{\implies{x=\frac{5}{2}}}$

$\sf{Substitute \ x=\frac{5}{2} \ in \ equation \ (1)}$

$\sf{\frac{5}{2}+y=\frac{7}{2}}$

$\sf{y=\frac{7}{2}-\frac{5}{2}}$

$\sf{y=\frac{7-5}{2}}$

$\sf{y=\frac{2}{2}}$

$\sf{\implies{y=1}}$

____________________________

$\sf{\implies{x^{2}-y^{2}}}$

$\sf{\implies{(\frac{5}{2})^{2}+1^{2}}}$

$\sf{\implies{\frac{25}{4}+1}}$

$\sf{\implies{\frac{25+4}{4}}}$

$\sf{\implies{\frac{29}{4}}}$

$\sf\purple{\tt{\therefore{Value \ of \ x^{2}-y^{2} \ is \ \frac{29}{4}}}}$